Chapter 14. Appendix

14.1. Practice Final Exam 1*

Questions 1-2 refer to the following:

An experiment consists of tossing two 12-sided dice (the numbers 1-12 are printed on the sides of each dice).

  • Let Event A = both dice show an even number

  • Let Event B = both dice show a number more than 8

Exercise 14.1.1. (Go to Solution)

Events A and B  are:

A. Mutually exclusive.
B. Independent.
C. Mutually exclusive and independent.
D. Neither mutually exclusive nor independent.

Exercise 14.1.2. (Go to Solution)

Find P ( A | B )

A.
B.
C.
C.

Exercise 14.1.3. (Go to Solution)

Which of the following are TRUE when we perform a hypothesis test on matched or paired samples?

A. Sample sizes are almost never small.
B. Two measurements are drawn from the same pair of individuals or objects.
C. Two sample averages are compared to each other.
D. Answer choices B and C are both true.

Questions 4 - 5 refer to the following:

118 students were asked what type of color their bedrooms were painted: light colors, dark colors or vibrant colors. The results were tabulated according to gender.

Table 14.1.
 Light colorsDark colorsVibrant colors
Female202228
Male10308

Exercise 14.1.4. (Go to Solution)

Find the probability that a randomly chosen student is male or has a bedroom painted with light colors.

A.
B.
C.
D.


Exercise 14.1.5. (Go to Solution)

Find the probability that a randomly chosen student is male given the student’s bedroom is painted with dark colors.

A.
B.
C.
D.

Questions 6 – 7 refer to the following:

We are interested in the number of times a teenager must be reminded to do his/her chores each week. A survey of 40 mothers was conducted. The table below shows the results of the survey.

Table 14.2.
X P ( x )
0
1
2 
3
4
5

Exercise 14.1.6. (Go to Solution)

Find the probability that a teenager is reminded 2 times.

A. 8
B.
C.
D. 2

Exercise 14.1.7. (Go to Solution)

Find the expected number of times a teenager is reminded to do his/her chores.

A. 15
B. 2.78
C. 1.0
D. 3.13

Questions 8 – 9 refer to the following:

On any given day, approximately 37.5% of the cars parked in the De Anza parking structure are parked crookedly. (Survey done by Kathy Plum.) We randomly survey 22 cars. We are interested in the number of cars that are parked crookedly.

Exercise 14.1.8. (Go to Solution)

For every 22 cars, how many would you expect to be parked crookedly, on average?

A. 8.25
B. 11
C. 18
D. 7.5

Exercise 14.1.9. (Go to Solution)

What is the probability that at least 10 of the 22 cars are parked crookedly.

A. 0.1263
B. 0.1607
C. 0.2870
D. 0.8393

Exercise 14.1.10. (Go to Solution)

Using a sample of 15 Stanford-Binet IQ scores, we wish to conduct a hypothesis test. Our claim is that the average IQ score on the Stanford-Binet IQ test is more than 100. It is known that the standard deviation of all Stanford-Binet IQ scores is 15 points. The correct distribution to use for the hypothesis test is:

A. Binomial
B. Student-t
C. Normal
D. Uniform

Questions 11 – 13 refer to the following:

De Anza College keeps statistics on the pass rate of students who enroll in math classes. In a sample of 1795 students enrolled in Math 1A (1st quarter calculus), 1428 passed the course. In a sample of 856 students enrolled in Math 1B (2nd quarter calculus), 662 passed. In general, are the pass rates of Math 1A and Math 1B statistically the same? Let A = the subscript for Math 1A and B = the subscript for Math 1B.

Exercise 14.1.11. (Go to Solution)

If you were to conduct an appropriate hypothesis test, the alternate hypothesis would be:

A. H a : p A = p B
B. H a : p A > p B
C. H o : p A = p B
D. H a : p A p B

Exercise 14.1.12. (Go to Solution)

The Type I error is to:

A. believe that the pass rate for Math 1A is the same as the pass rate for Math 1B when, in fact, the pass rates are different.
B. believe that the pass rate for Math 1A is different than the pass rate for Math 1B when, in fact, the pass rates are the same.
C. believe that the pass rate for Math 1A is greater than the pass rate for Math 1B when, in fact, the pass rate for Math 1A is less than the pass rate for Math 1B.
D. believe that the pass rate for Math 1A is the same as the pass rate for Math 1B when, in fact, they are the same.

Exercise 14.1.13. (Go to Solution)

The correct decision is to:

A. reject H o
B. not reject H o
C. not make a decision because of lack of information

Kia, Alejandra, and Iris are runners on the track teams at three different schools. Their running times, in minutes, and the statistics for the track teams at their respective schools, for a one mile run, are given in the table below:

Table 14.3.
 Running TimeSchool Average Running TimeSchool Standard Deviation
Kia4.95.2.15
Alejandra4.24.6.25
Iris4.54.9.12

Exercise 14.1.14. (Go to Solution)

Which student is the BEST when compared to the other runners at her school?

A. Kia
B. Alejandra
C. Iris
D. Impossible to determine

Questions 15 – 16 refer to the following:

The following adult ski sweater prices are from the Gorsuch Ltd. Winter catalog:

{ $212 ,$292 ,$278 ,$199 $280 , $236 }

Assume the underlying sweater price population is approximately normal. The null hypothesis is that the average price of adult ski sweaters from Gorsuch Ltd. is at least $275.

Exercise 14.1.15. (Go to Solution)

The correct distribution to use for the hypothesis test is:

A. Normal
B. Binomial
C. Student-t
D. Exponential

Exercise 14.1.16. (Go to Solution)

The hypothesis test:

A. is two-tailed
B. is left-tailed
C. is right-tailed
D. has no tails

Exercise 14.1.17. (Go to Solution)

Sara, a statistics student, wanted to determine the average number of books that college professors have in their office. She randomly selected 2 buildings on campus and asked each professor in the selected buildings how many books are in his/her office. Sara surveyed 25 professors. The type of sampling selected is a:

A. simple random sampling
B. systematic sampling
C. cluster sampling
D. stratified sampling

Exercise 14.1.18. (Go to Solution)

A clothing store would use which measure of the center of data when placing orders?

A. Mean
B. Median
C. Mode
D. IQR

Exercise 14.1.19. (Go to Solution)

In a hypothesis test, the p-value is

A. the probability that an outcome of the data will happen purely by chance when the null hypothesis is true.
B. called the preconceived alpha.
C. compared to beta to decide whether to reject or not reject the null hypothesis.
D. Answer choices A and B are both true.

Questions 20 - 22 refer to the following:

A community college offers classes 6 days a week: Monday through Saturday. Maria conducted a study of the students in her classes to determine how many days per week the students who are in her classes come to campus for classes. In each of her 5 classes she randomly selected 10 students and asked them how many days they come to campus for classes. The results of her survey are summarized in the table below.

Table 14.4.
Number of Days on CampusFrequencyRelative FrequencyCumulative Relative Frequency
12  
212.24 
310.20 
4  .98
50  
61.021.00

Exercise 14.1.20. (Go to Solution)

Combined with convenience sampling, what other sampling technique did Maria use?

A. simple random
B. systematic
C. cluster
D. stratified

Exercise 14.1.21. (Go to Solution)

How many students come to campus for classes 4 days a week?

A. 49
B. 25
C. 30
D. 13

Exercise 14.1.22. (Go to Solution)

What is the 60th percentile for the this data?

A. 2
B. 3
C. 4
D. 5

The next two questions refer to the following:

The following data are the results of a random survey of 110 Reservists called to active duty to increase security at California airports.

Table 14.5.
Number of DependentsFrequency
011
127
233
320
419

Exercise 14.1.23. (Go to Solution)

Construct a 95% Confidence Interval for the true population average number of dependents of Reservists called to active duty to increase security at California airports.

A. (1.85, 2.32)
B. (1.80, 2.36)
C. (1.97, 2.46)
D. (1.92, 2.50)


Exercise 14.1.24. (Go to Solution)

The 95% confidence Interval above means:

A. 5% of Confidence Intervals constructed this way will not contain the true population aveage number of dependents.
B. We are 95% confident the true population average number of dependents falls in the interval.
C. Both of the above answer choices are correct.
D. None of the above.

Exercise 14.1.25. (Go to Solution)

X~ U ( 4 , 10 ) . Find the 30th percentile.

A. 0.3000
B. 3
C. 5.8
D. 6.1

Exercise 14.1.26. (Go to Solution)

If X~ Exp ( 0.8 ) , then P ( X<μ )  =

A. 0.3679
B. 0.4727
C. 0.6321
D. cannot be determined

Exercise 14.1.27. (Go to Solution)

The lifetime of a computer circuit board is normally distributed with a mean of 2500 hours and a standard deviation of 60 hours. What is the probability that a randomly chosen board will last at most 2560 hours?

A. 0.8413
B. 0.1587
C. 0.3461
D. 0.6539

Exercise 14.1.28. (Go to Solution)

A survey of 123 Reservists called to active duty as a result of the September 11, 2001, attacks was conducted to determine the proportion that were married. Eighty-six reported being married. Construct a 98% confidence interval for the true population proportion of reservists called to active duty that are married.

A. (0.6030, 0.7954)
B. (0.6181, 0.7802)
C. (0.5927, 0.8057)
D. (0.6312, 0.7672)

Exercise 14.1.29. (Go to Solution)

Winning times in 26 mile marathons run by world class runners average 145 minutes with a standard deviation of 14 minutes. A sample of the last 10 marathon winning times is collected.

Let = average winning times for 10 marathons.

The distribution for  is:

A.
B. N (145, 14)
C. t 9
D. t 10

Exercise 14.1.30. (Go to Solution)

Suppose that Phi Beta Kappa honors the top 1% of college and university seniors. Assume that grade point averages (G.P.A.) at a certain college are normally distributed with a 2.5 average and a standard deviation of 0.5. What would be the minimum G.P.A. needed to become a member of Phi Beta Kappa at that college?

A. 3.99
B. 1.34
C. 3.00
D. 3.66

The number of people living on American farms has declined steadily during this century. Here are data on the farm population (in millions of persons) from 1935 to 1980.

Table 14.6.
Year1935194019451950195519601965197019751980
Population32.130.524.423.019.115.612.49.78.97.2

The linear regression equation is y-hat = 1166.93 – 0.5868x

Exercise 14.1.31. (Go to Solution)

What was the expected farm population (in millions of persons) for 1980?

A. 7.2
B. 5.1
C. 6.0
D. 8.0

Exercise 14.1.32. (Go to Solution)

In linear regression, which is the best possible SSE?

A. 13.46
B. 18.22
C. 24.05
D. 16.33

Exercise 14.1.33. (Go to Solution)

In regression analysis, if the correlation coefficient is close to 1 what can be said about the best fit line?

A. It is a horizontal line. Therefore, we can not use it.
B. There is a strong linear pattern. Therefore, it is most likely a good model to be used.
C. The coefficient correlation is close to the limit. Therefore, it is hard to make a decision.
D. We do not have the equation. Therefore, we can not say anything about it.

Question 34-36 refer to the following:

A study of the career plans of young women and men sent questionnaires to all 722 members of the senior class in the College of Business Administration at the University of Illinois. One question asked which major within the business program the student had chosen. Here are the data from the students who responded.

Table 14.7. Does the data suggest that there is a relationship between the gender of students and their choice of major?
 FemaleMale
Accounting6856
Administration9140
Ecomonics56
Finance6159

Exercise 14.1.34. (Go to Solution)

The distribution for the test is:

A. Chi2 8
B. Chi2 3
C. t 722
D. N ( 0 ,1 )

Exercise 14.1.35. (Go to Solution)

The expected number of female who choose Finance is :

A. 37
B. 61
C. 60
D. 70

Exercise 14.1.36. (Go to Solution)

The p-value is 0.0127. The conclusion to the test is:

A. The choice of major and the gender of the student are independent of each other.
B. The choice of major and the gender of the student are not independent of each other.
C. Students find Economics very hard.
D. More females prefer Administration than males.

Exercise 14.1.37. (Go to Solution)

An agency reported that the work force nationwide is composed of 10% professional, 10% clerical, 30% skilled, 15% service, and 35% semiskilled laborers. A random sample of 100 San Jose residents indicated 15 professional, 15 clerical, 40 skilled, 10 service, and 20 semiskilled laborers. At α = .10 does the work force in San Jose appear to be consistent with the agency report for the nation? Which kind of test is it?

A. Chi2 goodness of fit
B. Chi2 test of independence
C. Independent groups proportions
D. Unable to determine

Solutions to Exercises

Solution to Exercise 14.1.1. (Return to Exercise)

B: Independent.


Solution to Exercise 14.1.2. (Return to Exercise)

C:


Solution to Exercise 14.1.3. (Return to Exercise)

B: Two measurements are drawn from the same pair of individuals or objects.


Solution to Exercise 14.1.4. (Return to Exercise)

B:


Solution to Exercise 14.1.5. (Return to Exercise)

D:


Solution to Exercise 14.1.6. (Return to Exercise)

B:


Solution to Exercise 14.1.7. (Return to Exercise)

B: 2.78


Solution to Exercise 14.1.8. (Return to Exercise)

A: 8.25


Solution to Exercise 14.1.9. (Return to Exercise)

C: 0.2870


Solution to Exercise 14.1.10. (Return to Exercise)

C: Normal


Solution to Exercise 14.1.11. (Return to Exercise)

D: H a : p A p B


Solution to Exercise 14.1.12. (Return to Exercise)

B: believe that the pass rate for Math 1A is different than the pass rate for Math 1B when, in fact, the pass rates are the same.


Solution to Exercise 14.1.13. (Return to Exercise)

B: not reject H o


Solution to Exercise 14.1.14. (Return to Exercise)

C: Iris


Solution to Exercise 14.1.15. (Return to Exercise)

C: Student-t


Solution to Exercise 14.1.16. (Return to Exercise)

B: is left-tailed


Solution to Exercise 14.1.17. (Return to Exercise)

C: cluster sampling


Solution to Exercise 14.1.18. (Return to Exercise)

C: Mode


Solution to Exercise 14.1.19. (Return to Exercise)

A: the probability that an outcome of the data will happen purely by chance when the null hypothesis is true.


Solution to Exercise 14.1.20. (Return to Exercise)

D: stratified


Solution to Exercise 14.1.21. (Return to Exercise)

B: 25


Solution to Exercise 14.1.22. (Return to Exercise)

C: 4


Solution to Exercise 14.1.23. (Return to Exercise)

A: (1.85, 2.32)


Solution to Exercise 14.1.24. (Return to Exercise)

C: Both above are correct.


Solution to Exercise 14.1.25. (Return to Exercise)

C: 5.8


Solution to Exercise 14.1.26. (Return to Exercise)

C: 0.6321


Solution to Exercise 14.1.27. (Return to Exercise)

A: 0.8413


Solution to Exercise 14.1.28. (Return to Exercise)

A: (0.6030, 0.7954)


Solution to Exercise 14.1.29. (Return to Exercise)

A:


Solution to Exercise 14.1.30. (Return to Exercise)

D: 3.66


Solution to Exercise 14.1.31. (Return to Exercise)

B: 5.1


Solution to Exercise 14.1.32. (Return to Exercise)

A: 13.46


Solution to Exercise 14.1.33. (Return to Exercise)

B: There is a strong linear pattern. Therefore, it is most likely a good model to be used.


Solution to Exercise 14.1.34. (Return to Exercise)

B: Chi2 3


Solution to Exercise 14.1.35. (Return to Exercise)

D: 70


Solution to Exercise 14.1.36. (Return to Exercise)

B: The choice of major and the gender of the student are not independent of each other.


Solution to Exercise 14.1.37. (Return to Exercise)

A: Chi2 goodness of fit


14.2. Practice Final Exam 2*

Exercise 14.2.1. (Go to Solution)

A study was done to determine the proportion of teenagers that own a car. The true proportion of teenagers that own a car is the:

A. statistic
B. parameter
C. population
D. variable


The next two questions refer to the following data:

Table 14.8.
valuefrequency
01
14
27
39
64

Exercise 14.2.2. (Go to Solution)

The box plot for the data is:

A. atat1.png
B. atat2.png
C. atat3.png
D. atat4.png


Exercise 14.2.3. (Go to Solution)

If 6 were added to each value of the data in the table, the 15th percentile of the new list of values is:

A. 6
B. 1
C. 7
D. 8

The next two questions refer to the following situation:

Suppose that the probability of a drought in any independent year is 20%. Out of those years in which a drought occurs, the probability of water rationing is 10%. However, in any year, the probability of water rationing is 5%.

Exercise 14.2.4. (Go to Solution)

What is the probability of both a drought and water rationing occurring?

A. 0.05
B. 0.01
C. 0.02
D. 0.30

Exercise 14.2.5. (Go to Solution)

Which of the following is true?

A. drought and water rationing are independent events
B. drought and water rationing are mutually exclusive events
C. none of the above

The next two questions refer to the following situation:

Suppose that a survey yielded the following data:

Table 14.9. Favorite Pie Type
genderapplepumpkinpecan
female401030
male203010

Exercise 14.2.6. (Go to Solution)

Suppose that one individual is randomly chosen. The probability that the person’s favorite pie is apple or the person is male is:

A.
B.
C.
D.

Exercise 14.2.7. (Go to Solution)

Suppose H o is: Favorite pie type and gender are independent.

The p-value is:

A. ≈ 0
B. 1
C. 0.05
D. cannot be determined

The next two questions refer to the following situation:

Let’s say that the probability that an adult watches the news at least once per week is 0.60. We randomly survey 14 people. Of interest is the number that watch the news at least once per week.

Exercise 14.2.8. (Go to Solution)

Which of the following statements is FALSE?

A. X ~ B (14, 0.60)
B. The values for x are: { 1 ,2 ,3 ,… , 14 }
C. μ = 8.4
D. P ( X = 5 ) = 0.0408

Exercise 14.2.9. (Go to Solution)

Find the probability that at least 6 adults watch the news.

A.
B. 0.8499
C. 0.9417
D. 0.6429

Exercise 14.2.10. (Go to Solution)

The following histogram is most likely to be a result of sampling from which distribution?

desc_stats_final2_2.png

A. Chi-Square
B. Exponential
C. Uniform
D. Binomial

The ages of campus day and evening students is known to be normally distributed. A sample of 6 campus day and evening students reported their ages (in years) as: { 18 ,35 ,27 ,45 , 20 , 20 }

Exercise 14.2.11. (Go to Solution)

What is the error bound for the 90% confidence interval of the true average age?

A. 11.2
B. 22.3
C. 17.5
D. 8.7

Exercise 14.2.12. (Go to Solution)

If a normally distributed random variable has µ = 0 and σ = 1 , then 97.5% of the population values lie above:

A. -1.96
B. 1.96
C. 1
D. -1

The next three questions refer to the following situation:

The amount of money a customer spends in one trip to the supermarket is known to have an exponential distribution. Suppose the average amount of money a customer spends in one trip to the supermarket is $72.

Exercise 14.2.13. (Go to Solution)

What is the probability that one customer spends less than $72 in one trip to the supermarket?

A. 0.6321
B. 0.5000
C. 0.3714
D. 1

Exercise 14.2.14. (Go to Solution)

How much money altogether would you expect next 5 customers to spend in one trip to the supermarket (in dollars)?

A. 72
B.
C. 5184
D. 360

Exercise 14.2.15. (Go to Solution)

If you want to find the probability that the average of 5 customers is less than $60, the distribution to use is:

A. N ( 72 , 72 )
B.
C. Exp ( 72 )
D.

The next three questions refer to the following situation:

The amount of time it takes a fourth grader to carry out the trash is uniformly distributed in the interval from 1 to 10 minutes.

Exercise 14.2.16. (Go to Solution)

What is the probability that a randomly chosen fourth grader takes more than 7 minutes to take out the trash?

A.
B.
C.
D.

Exercise 14.2.17. (Go to Solution)

Which graph best shows the probability that a randomly chosen fourth grader takes more than 6 minutes to take out the trash given that he/she has already taken more than 3 minutes?

desc_stats_final2_3.png


Exercise 14.2.18. (Go to Solution)

We should expect a fourth grader to take how many minutes to take out the trash?

A. 4.5
B. 5.5
C. 5
D. 10

The next three questions refer to the following situation:

At the beginning of the quarter, the amount of time a student waits in line at the campus cafeteria is normally distributed with a mean of 5 minutes and a standard deviation of 1.5 minutes.

Exercise 14.2.19. (Go to Solution)

What is the 90th percentile of waiting times (in minutes)?

A. 1.28
B. 90
C. 7.47
D. 6.92

Exercise 14.2.20. (Go to Solution)

The median waiting time (in minutes) for one student is:

A. 5
B. 50
C. 2.5
D. 1.5

Exercise 14.2.21. (Go to Solution)

Find the probability that the average wait time of 10 students is at most 5.5 minutes.

A. 0.6301
B. 0.8541
C. 0.3694
D. 0.1459

Exercise 14.2.22. (Go to Solution)

A sample of 80 software engineers in Silicon Valley is taken and it is found that 20% of them earn approximately $50,000 per year. A point estimate for the true proportion of engineers in Silicon Valley who earn $50,000 per year is:

A. 16
B. 0.2
C. 1
D. 0.95

Exercise 14.2.23. (Go to Solution)

If . 1587 where Z ~ N ( 0, 1 ) , then α is equal to:

A. -1
B. 0.1587
C. 0.8413
D. 1

Exercise 14.2.24. (Go to Solution)

A professor tested 35 students to determine their entering skills. At the end of the term, after completing the course, the same test was administered to the same 35 students to study their improvement. This would be a test of:

A. independent groups
B. 2 proportions
C. dependent groups
D. exclusive groups

Exercise 14.2.25. (Go to Solution)

A math exam was given to all the third grade children attending ABC School. Two random samples of scores were taken.

Table 14.10.
 n s
Boys55825
Girls60867

Which of the following correctly describes the results of a hypothesis test of the claim, “There is a difference between the mean scores obtained by third grade girls and boys at the 5 % level of significance”?

A. Do not reject H o . There is no difference in the mean scores.
B. Do not reject H o . There is a difference in the mean scores.
C. Reject H o . There is no difference in the mean scores.
D. Reject H o . There is a difference in the mean scores.

Exercise 14.2.26. (Go to Solution)

In a survey of 80 males, 45 had played an organized sport growing up. Of the 70 females surveyed, 25 had played an organized sport growing up. We are interested in whether the proportion for males is higher than the proportion for females. The correct conclusion is:

A. The proportion for males is the same as the proportion for females.
B. The proportion for males is not the same as the proportion for females.
C. The proportion for males is higher than the proportion for females.
D. Not enough information to determine.

Exercise 14.2.27. (Go to Solution)

Note: Chi-Square Test of a Single Variance; Not all classes cover this topic. From past experience, a statistics teacher has found that the average score on a midterm is 81 with a standard deviation of 5.2. This term, a class of 49 students had a standard deviation of 5 on the midterm. Do the data indicate that we should reject the teacher’s claim that the standard deviation is 5.2? Use α = 0.05 .

A. Yes
B. No
C. Not enough information given to solve the problem

Exercise 14.2.28. (Go to Solution)

Note: F Distribution Test of ANOVA; Not all classes cover this topic. Three loading machines are being compared. Ten samples were taken for each machine. Machine I took an average of 31 minutes to load packages with a standard deviation of 2 minutes. Machine II took an average of 28 minutes to load packages with a standard deviation of 1.5 minutes. Machine III took an average of 29 minutes to load packages with a standard deviation of 1 minute. Find the p-value when testing that the average loading times are the same.

A. the p–value is close to 0
B. p–value is close to 1
C. Not enough information given to solve the problem

The next three questions refer to the following situation:

A corporation has offices in different parts of the country. It has gathered the following information concerning the number of bathrooms and the number of employees at seven sites:

Table 14.11.
Number of employees x6507308109001021071150
Number of bathrooms y4050546182110121

Exercise 14.2.29. (Go to Solution)

Is the correlation between the number of employees and the number of bathrooms significant?

A. Yes
B. No
C. Not enough information to answer question

Exercise 14.2.30. (Go to Solution)

The linear regression equation is:

A.
B.
C.
D.

Exercise 14.2.31. (Go to Solution)

If a site has 1150 employees, approximately how many bathrooms should it have?

A. 69
B. 91
C. 91,954
D. We should not be estimating here.

Exercise 14.2.32. (Go to Solution)

Note: Chi-Square Test of a Single Variance; Not all classes cover this topic. Suppose that a sample of size 10 was collected, with = 4.4 and s = 1.4 .

H o : σ 2 = 1.6 vs. H a : σ 2 ≠ 1.6. Which graph best describes the results of the test?

desc_stats_final2_4.png


Exercise 14.2.33. (Go to Solution)

64 backpackers were asked the number of days their latest backpacking trip was. The number of days is given in the table below:

Table 14.12.
# of days12345678
Frequency59612710510

Conduct an appropriate test to determine if the distribution is uniform.

A. The p–value is > 0.10 , the distribution is uniform.
B. The p–value is < 0.01 , the distribution is uniform.
C. The p–value is between 0.01 and 0.10, but without α there is not enough information
D. There is no such test that can be conducted.

Exercise 14.2.34. (Go to Solution)

­Note: F Distribution test of ANOVA; Not all classes cover this topic. Which of the following statements is true when using one-way ANOVA?

A. The populations from which the samples are selected have different distributions.
B. The sample sizes are large.
C. The test is to determine if the different groups have the same averages.
D. There is a correlation between the factors of the experiment.

Solutions to Exercises

Solution to Exercise 14.2.1. (Return to Exercise)

B: parameter


Solution to Exercise 14.2.2. (Return to Exercise)

A


Solution to Exercise 14.2.3. (Return to Exercise)

C: 7


Solution to Exercise 14.2.4. (Return to Exercise)

C: 0.02


Solution to Exercise 14.2.5. (Return to Exercise)

C: none of the above


Solution to Exercise 14.2.6. (Return to Exercise)

D:


Solution to Exercise 14.2.7. (Return to Exercise)

A: ≈ 0


Solution to Exercise 14.2.8. (Return to Exercise)

B: The values for x are: { 1 ,2 ,3 ,… , 14 }


Solution to Exercise 14.2.9. (Return to Exercise)

C: 0.9417


Solution to Exercise 14.2.10. (Return to Exercise)

D: Binomial


Solution to Exercise 14.2.11. (Return to Exercise)

D: 8.7


Solution to Exercise 14.2.12. (Return to Exercise)

A: -1.96


Solution to Exercise 14.2.13. (Return to Exercise)

A: 0.6321


Solution to Exercise 14.2.14. (Return to Exercise)

D: 360


Solution to Exercise 14.2.15. (Return to Exercise)

B:


Solution to Exercise 14.2.16. (Return to Exercise)

A:


Solution to Exercise 14.2.17. (Return to Exercise)

(D)


Solution to Exercise 14.2.18. (Return to Exercise)

B: 5.5


Solution to Exercise 14.2.19. (Return to Exercise)

D: 6.92


Solution to Exercise 14.2.20. (Return to Exercise)

A: 5


Solution to Exercise 14.2.21. (Return to Exercise)

B: 0.8541


Solution to Exercise 14.2.22. (Return to Exercise)

B: 0.2


Solution to Exercise 14.2.23. (Return to Exercise)

A: -1


Solution to Exercise 14.2.24. (Return to Exercise)

C: dependent groups


Solution to Exercise 14.2.25. (Return to Exercise)

D: Reject H o . There is a difference in the mean scores.


Solution to Exercise 14.2.26. (Return to Exercise)

C: The proportion for males is higher than the proportion for females.


Solution to Exercise 14.2.27. (Return to Exercise)

B: No


Solution to Exercise 14.2.28. (Return to Exercise)

B: p-value is close to 1.


Solution to Exercise 14.2.29. (Return to Exercise)

B: No


Solution to Exercise 14.2.30. (Return to Exercise)

C:


Solution to Exercise 14.2.31. (Return to Exercise)

D: We should not be estimating here.


Solution to Exercise 14.2.32. (Return to Exercise)

A


Solution to Exercise 14.2.33. (Return to Exercise)

B: The p–value is < 0.01 , the distribution is uniform.


Solution to Exercise 14.2.34. (Return to Exercise)

C: The test is to determine if the different groups have the same averages.


14.3. Data Sets*

Lap Times

The following tables provide lap times from Terri Vogel’s Log Book. Times are recorded in seconds for 2.5-mile laps completed in a series of races and practice runs.

Table 14.13. Race Lap Times (in Seconds)
 Lap 1Lap 2Lap 3Lap 4Lap 5Lap 6Lap 7
Race 1135130131132130131133
Race 2134131131129128128129
Race 3129128127127130127129
Race 4125125126125124125125
Race 5133132132132131130132
Race 6130130130129129130129
Race 7132131133131134134131
Race 8127128127130128126128
Race 9132130127128126127124
Race 10135131131132130131130
Race 11132131132131130129129
Race 12134130130130131130130
Race 13128127128128128129128
Race 14132131131131132130130
Race 15136129129129129129129
Race 16129129129128128129129
Race 17134131132131132132132
Race 18129129130130133133127
Race 19130129129129129129128
Race 20131128130128129130130
Table 14.14. Practice Lap Times (in Seconds)
 Lap 1Lap 2Lap 3Lap 4Lap 5Lap 6Lap 7
Practice 1142143180137134134172
Practice 2140135134133128128131
Practice 3130133130128135133133
Practice 4141136137136136136145
Practice 5140138136137135134134
Practice 6142142139138129129127
Practice 7139137135135137134135
Practice 8143136134133134133132
Practice 9135134133133132132133
Practice 10131130128129127128127
Practice 11143139139138138137138
Practice 12132133131129128127126
Practice 13149144144139138138137
Practice 14133132137133134130131
Practice 15138136133133132131131

Stock Prices

The following table lists initial public offering (IPO) stock prices for all 1999 stocks that at least doubled in value during the first day of trading.

Table 14.15. IPO Offer Prices
$17.00$23.00$14.00$16.00$12.00$26.00
$20.00$22.00$14.00$15.00$22.00$18.00
$18.00$21.00$21.00$19.00$15.00$21.00
$18.00$17.00$15.00$25.00$14.00$30.00
$16.00$10.00$20.00$12.00$16.00$17.44
$16.00$14.00$15.00$20.00$20.00$16.00
$17.00$16.00$15.00$15.00$19.00$48.00
$16.00$18.00$9.00$18.00$18.00$20.00
$8.00$20.00$17.00$14.00$11.00$16.00
$19.00$15.00$21.00$12.00$8.00$16.00
$13.00$14.00$15.00$14.00$13.41$28.00
$21.00$17.00$28.00$17.00$19.00$16.00
$17.00$19.00$18.00$17.00$15.00 
$14.00$21.00$12.00$18.00$24.00 
$15.00$23.00$14.00$16.00$12.00 
$24.00$20.00$14.00$14.00$15.00 
$14.00$19.00$16.00$38.00$20.00 
$24.00$16.00$8.00$18.00$17.00 
$16.00$15.00$7.00$19.00$12.00 
$8.00$23.00$12.00$18.00$20.00 
$21.00$34.00$16.00$26.00$14.00 

Note

Data compiled by Jay R. Ritter of Univ. of Florida using data from Securities Data Co. and Bloomberg.

14.4. Group Projects

Group Project: Univariate Data*

Student Learning Objectives

  • The student will design and carry out a survey.

  • The student will analyze and graphically display the results of the survey.

Instructions

As you complete each task below, check it off. Answer all questions in your summary.

____ Decide what data you are going to study.

Examples

Here are two examples, but you may NOT use them: number of M&M’s per small bag, number of pencils students have in their backpacks.

____ Are your data discrete or continuous? How do you know?
____ Decide how you are going to collect the data (for instance, buy 30 bags of M&M’s; collect data from the World Wide Web).
____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random (using a random number generator) sampling. Do not use convenience sampling. What method did you use? Why did you pick that method?
____ Conduct your survey. Your data size must be at least 30.
____ Summarize your data in a chart with columns showing data value, frequency, relative frequency and cumulative relative frequency.
____ Answer the following (rounded to 2 decimal places):
1. =
2. s =
3. First quartile =
4. Median =
5. 70th percentile =
____ What value is 2 standard deviations above the mean?
____ What value is 1.5 standard deviations below the mean?
____ Construct a histogram displaying your data.
____ In complete sentences, describe the shape of your graph.
____ Do you notice any potential outliers? If so, what values are they? Show your work in how you used the potential outlier formula in Chapter 2 (since you have univariate data) to determine whether or not the values might be outliers.
____ Construct a box plot displaying your data.
____ Does the middle 50% of the data appear to be concentrated together or spread apart? Explain how you determined this.
____ Looking at both the histogram and the box plot, discuss the distribution of your data.

Assignment Checklist

You need to turn in the following typed and stapled packet, with pages in the following order:

____ Cover sheet: name, class time, and name of your study
____ Summary page: This should contain paragraphs written with complete sentences. It should include answers to all the questions above. It should also include statements describing the population under study, the sample, a parameter or parameters being studied, and the statistic or statistics produced.
____ URL for data, if your data are from the World Wide Web.
____ Chart of data, frequency, relative frequency and cumulative relative frequency.
____ Page(s) of graphs: histogram and box plot.

Group Project: Continuous Distributions and Central Limit Theorem*

Student Learning Objectives

  • The student will collect a sample of continuous data.

  • The student will attempt to fit the data sample to various distribution models.

  • The student will validate the Central Limit Theorem.

Instructions

As you complete each task below, check it off. Answer all questions in your summary.

Part I: Sampling

____ Decide what continuous data you are going to study. (Here are two examples, but you may NOT use them: the amount of money a student spends on college supplies this term or the length of a long distance telephone call.)
____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random (using a random number generator) sampling. Do not use convenience sampling. What method did you use? Why did you pick that method?
____ Conduct your survey. Gather at least 150 pieces of continuous quantitative data.
____ Define (in words) the random variable for your data. X = _______
____ Create 2 lists of your data: (1) unordered data, (2) in order of smallest to largest.
____ Find the sample mean and the sample standard deviation (rounded to 2 decimal places).
1. =
2. s =
____ Construct a histogram of your data containing 5 - 10 intervals of equal width. The histogram should be a representative display of your data. Label and scale it.

Part II: Possible Distributions

____ Suppose that X followed the theoretical distributions below. Set up each distribution using the appropriate information from your data.
____ Uniform: X ~ U ____________ Use the lowest and highest values as a and b .
____ Exponential: X ~ Exp ____________Use to estimate μ .
____ Normal: X ~ N ____________ Use to estimate for μ and s to estimate for σ .
____ Must your data fit one of the above distributions? Explain why or why not.
____ Could the data fit 2 or 3 of the above distributions (at the same time)? Explain.
____ Calculate the value k (an X value) that is 1.75 standard deviations above the sample mean. k = _________ (rounded to 2 decimal places) Note:
____ Determine the relative frequencies (RF) rounded to 4 decimal places.
1.
2. RF(X < k) =
3. RF(X > k) =
4. RF(X = k) =

Use a separate piece of paper for EACH distribution (uniform, exponential, normal) to respond to the following questions.

Note

You should have one page for the uniform, one page for the exponential, and one page for the normal

____ State the distribution: X ~ _________
____ Draw a graph for each of the three theoretical distributions. Label the axes and mark them appropriately.
____ Find the following theoretical probabilities (rounded to 4 decimal places).
1. P(X < k ) =
2. P(X > k ) =
3. P(X = k ) =
____ Compare the relative frequencies to the corresponding probabilities. Are the values close?
____ Does it appear that the data fit the distribution well? Justify your answer by comparing the probabilities to the relative frequencies, and the histograms to the theoretical graphs.

Part III: CLT Experiments

______ From your original data (before ordering), use a random number generator to pick 40 samples of size 5. For each sample, calculate the average.
______ On a separate page, attached to the summary, include the 40 samples of size 5, along with the 40 sample averages.
______ List the 40 averages in order from smallest to largest.
______ Define the random variable, , in words. =
______ State the approximate theoretical distribution of .
______ Base this on the mean and standard deviation from your original data.
______ Construct a histogram displaying your data. Use 5 to 6 intervals of equal width. Label and scale it.
Calculate the value (an value) that is 1.75 standard deviations above the sample mean. = _____ (rounded to 2 decimal places)
Determine the relative frequencies (RF) rounded to 4 decimal places.
1. RF( ) =
2. RF( > ) =
3. RF( = ) =
Find the following theoretical probabilities (rounded to 4 decimal places).
  • 1. P( < ) =

  • 2. P( > ) =

  • 3. P( = ) =

______ Draw the graph of the theoretical distribution of X .
______ Answer the questions below.
______ Compare the relative frequencies to the probabilities. Are the values close?
______ Does it appear that the data of averages fit the distribution of well? Justify your answer by comparing the probabilities to the relative frequencies, and the histogram to the theoretical graph.
______ In 3 - 5 complete sentences for each, answer the following questions. Give thoughtful explanations.
______ In summary, do your original data seem to fit the uniform, exponential, or normal distributions? Answer why or why not for each distribution. If the data do not fit any of those distributions, explain why.
______ What happened to the shape and distribution when you averaged your data? In theory, what should have happened? In theory, would “it” always happen? Why or why not?
______ Were the relative frequencies compared to the theoretical probabilities closer when comparing the X or distributions? Explain your answer.

Assignment Checklist

You need to turn in the following typed and stapled packet, with pages in the following order:

____ Cover sheet: name, class time, and name of your study
____ Summary pages: These should contain several paragraphs written with complete sentences that describe the experiment, including what you studied and your sampling technique, as well as answers to all of the questions above.
____ URL for data, if your data are from the World Wide Web.
____ Pages, one for each theoretical distribution, with the distribution stated, the graph, and the probability questions answered
____ Pages of the data requested
____ All graphs required

Partner Project: Hypothesis Testing - Article*

Student Learning Objectives

  • The student will identify a hypothesis testing problem in print.

  • The student will conduct a survey to verify or dispute the results of the hypothesis test.

  • The student will summarize the article, analysis, and conclusions in a report.

Instructions

As you complete each task below, check it off. Answer all questions in your summary.

____ Find an article in a newspaper, magazine or on the internet which makes a claim about ONE population mean or ONE population proportion. The claim may be based upon a survey that the article was reporting on. Decide whether this claim is the null or alternate hypothesis.
____ Copy or print out the article and include a copy in your project, along with the source.
____ State how you will collect your data. (Convenience sampling is not acceptable.)
____ Conduct your survey. You must have more than 50 responses in your sample. When you hand in your final project, attach the tally sheet or the packet of questionnaires that you used to collect data. Your data must be real.
____ State the statistics that are a result of your data collection: sample size, sample mean, and sample standard deviation, OR sample size and number of successes.
____ Make 2 copies of the appropriate solution sheet.
____ Record the hypothesis test on the solution sheet, based on your experiment. Do a DRAFT solution first on one of the solution sheets and check it over carefully. Have a classmate check your solution to see if it is done correctly. Make your decision using a 5% level of significance. Include the 95% confidence interval on the solution sheet.
____ Create a graph that illustrates your data. This may be a pie or bar chart or may be a histogram or box plot, depending on the nature of your data. Produce a graph that makes sense for your data and gives useful visual information about your data. You may need to look at several types of graphs before you decide which is the most appropriate for the type of data in your project.
____ Write your summary (in complete sentences and paragraphs, with proper grammar and correct spelling) that describes the project. The summary MUST include:
1. Brief discussion of the article, including the source.
2. Statement of the claim made in the article (one of the hypotheses).
3. Detailed description of how, where, and when you collected the data, including the sampling technique. Did you use cluster, stratified, systematic, or simple random sampling (using a random number generator)? As stated above, convenience sampling is not acceptable.
4. Conclusion about the article claim in light of your hypothesis test. This is the conclusion of your hypothesis test, stated in words, in the context of the situation in your project in sentence form, as if you were writing this conclusion for a non-statistician.
5. Sentence interpreting your confidence interval in the context of the situation in your project.

Assignment Checklist

Turn in the following typed (12 point) and stapled packet for your final project:

____ Cover sheet containing your name(s), class time, and the name of your study.
____ Summary, which includes all items listed on summary checklist.
____ Solution sheet neatly and completely filled out. The solution sheet does not need to be typed.
____ Graphic representation of your data, created following the guidelines discussed above. Include only graphs which are appropriate and useful.
____ Raw data collected AND a table summarizing the sample data (n, xbar and s; or x, n, and p’, as appropriate for your hypotheses). The raw data does not need to be typed, but the summary does. Hand in the data as you collected it. (Either attach your tally sheet or an envelope containing your questionnaires.)

Partner Project: Hypothesis Testing - Word Problem*

Student Learning Objectives

  • The student will write, edit, and solve a hypothesis testing word problem.

Instructions

Write an original hypothesis testing problem for either ONE population mean or ONE population proportion. As you complete each task, check it off. Answer all questions in your summary. Look at the homework for the Hypothesis Testing: Single Mean and Single Proportion chapter for examples (poems, two acts of a play, a work related problem). The problems with names attached to them are problems written by students in past quarters. Some other examples that are not in the homework include: a soccer hypothesis testing poster, a cartoon, a news reports, a children’s story, a song.

____ Your problem must be original and creative. It also must be in proper English. If English is difficult for you, have someone edit your problem.
____ Your problem must be at least ½ page, typed and singled spaced. This DOES NOT include the data. Data will make the problem longer and that is fine. For this problem, the data and story may be real or fictional.
____ In the narrative of the problem, make it very clear what the null and alternative hypotheses are.
____ Your sample size must be LARGER THAN 50 (even if it is fictional).
____ State in your problem how you will collect your data.
____ Include your data with your word problem.
____ State the statistics that are a result of your data collection: sample size, sample mean, and sample standard deviation, OR sample size and number of successes.
____ Create a graph that illustrates your problem. This may be a pie or bar chart or may be a histogram or box plot, depending on the nature of your data. Produce a graph that makes sense for your data and gives useful visual information about your data. You may need to look at several types of graphs before you decide which is the most appropriate for your problem.
____ Make 2 copies of the appropriate solution sheet.
____ Record the hypothesis test on the solution sheet, based on your problem. Do a DRAFT solution first on one of the solution sheets and check it over carefully. Make your decision using a 5% level of significance. Include the 95% confidence interval on the solution

Assignment Checklist

You need to turn in the following typed (12 point) and stapled packet for your final project:

____ Cover sheet containing your name, the name of your problem, and the date
____ The problem
____ Data for the problem
____ Solution sheet neatly and completely filled out. The solution sheet does not need to be typed.
____ Graphic representation of the data, created following the guidelines discussed above. Include only graphs that are appropriate and useful.
____ Sentences interpreting the results of the hypothesis test and the confidence interval in the context of the situation in the project.

Group Project: Bivariate Data, Linear Regression, and Univariate Data*

Student Learning Objectives

  • The students will collect a bivariate data sample through the use of appropriate sampling techniques.

  • The student will attempt to fit the data to a linear model.

  • The student will determine the appropriateness of linear fit of the model.

  • The student will analyze and graph univariate data.

Instructions

  1. As you complete each task below, check it off. Answer all questions in your introduction or summary.

  2. Check your course calendar for intermediate and final due dates.

  3. Graphs may be constructed by hand or by computer, unless your instructor informs you otherwise. All graphs must be neat and accurate.

  4. All other responses must be done on the computer.

  5. Neatness and quality of explanations are used to determine your final grade.

Part I: Bivariate Data

____ State the bivariate data your group is going to study.

Examples

Here are two examples, but you may NOT use them: height vs. weight and age vs. running distance.

____ Describe how your group is going to collect the data (for instance, collect data from the web, survey students on campus).
____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random sampling (using a random number generator) sampling. Convenience sampling is NOT acceptable.
____ Conduct your survey. Your number of pairs must be at least 30.
____ Print out a copy of your data.
____ On a separate sheet of paper construct a scatter plot of the data. Label and scale both axes.
____ State the least squares line and the correlation coefficient.
____ On your scatter plot, in a different color, construct the least squares line.
____ Is the correlation coefficient significant? Explain and show how you determined this.
____ Interpret the slope of the linear regression line in the context of the data in your project. Relate the explanation to your data, and quantify what the slope tells you.
____ Does the regression line seem to fit the data? Why or why not? If the data does not seem to be linear, explain if any other model seems to fit the data better.
____ Are there any outliers? If so, what are they? Show your work in how you used the potential outlier formula in the Linear Regression and Correlation chapter (since you have bivariate data) to determine whether or not any pairs might be outliers.

Part II: Univariate Data

In this section, you will use the data for ONE variable only. Pick the variable that is more interesting to analyze. For example: if your independent variable is sequential data such as year with 30 years and one piece of data per year, your x-values might be 1971, 1972, 1973, 1974, …, 2000. This would not be interesting to analyze. In that case, choose to use the dependent variable to analyze for this part of the project.

_____ Summarize your data in a chart with columns showing data value, frequency, relative frequency, and cumulative relative frequency.
_____ Answer the following, rounded to 2 decimal places:
1. Sample mean =
2. Sample standard deviation =
3. First quartile =
4. Third quartile =
5. Median =
6. 70th percentile =
7. Value that is 2 standard deviations above the mean =
8. Value that is 1.5 standard deviations below the mean =
_____ Construct a histogram displaying your data. Group your data into 6 – 10 intervals of equal width. Pick regularly spaced intervals that make sense in relation to your data. For example, do NOT group data by age as 20-26,27-33,34-40,41-47,48-54,55-61 … Instead, maybe use age groups 19.5-24.5, 24.5-29.5, … or 19.5-29.5, 29.5-39.5, 39.5-49.5, …
_____ In complete sentences, describe the shape of your histogram.
_____ Are there any potential outliers? Which values are they? Show your work and calculations as to how you used the potential outlier formula in chapter 2 (since you are now using univariate data) to determine which values might be outliers.
_____ Construct a box plot of your data.
_____ Does the middle 50% of your data appear to be concentrated together or spread out? Explain how you determined this.
_____ Looking at both the histogram AND the box plot, discuss the distribution of your data. For example: how does the spread of the middle 50% of your data compare to the spread of the rest of the data represented in the box plot; how does this correspond to your description of the shape of the histogram; how does the graphical display show any outliers you may have found; does the histogram show any gaps in the data that are not visible in the box plot; are there any interesting features of your data that you should point out.

Due Dates

  • Part I, Intro: __________ (keep a copy for your records)

  • Part I, Analysis: __________ (keep a copy for your records)

  • Entire Project, typed and stapled: __________

    ____ Cover sheet: names, class time, and name of your study.
    ____ Part I: label the sections “Intro” and “Analysis.”
    ____ Part II:
    ____ Summary page containing several paragraphs written in complete sentences describing the experiment, including what you studied and how you collected your data. The summary page should also include answers to ALL the questions asked above.
    ____ All graphs requested in the project.
    ____ All calculations requested to support questions in data.
    ____ Description: what you learned by doing this project, what challenges you had, how you overcame the challenges.

Note

Include answers to ALL questions asked, even if not explicitly repeated in the items above.

14.5. Solution Sheets

Solution Sheet: Hypothesis Testing for Single Mean and Single Proportion*

Class Time:

Name:

a. H o :
b. H a :
c. In words, CLEARLY state what your random variable or P’ represents.
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p -value? In 1 – 2 complete sentences, explain what the p -value means for this problem.
g. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p -value.

Figure 14.1. 

Figure (wave.PNG)

h. Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
i. Alpha:
ii. Decision:
iii. Reason for decision:
iv. Conclusion:
i. Construct a 95% Confidence Interval for the true mean or proportion. Include a sketch of the graph of the situation. Label the point estimate and the lower and upper bounds of the Confidence Interval.

Figure 14.2. 

Figure (wave.PNG)

Solution Sheet: Hypothesis Testing for Two Means, Paired Data, and Two Proportions*

Class Time:

Name:

a. H o : _______
b. H a : _______
c. In words, clearly state what your random variable , P 1 P 2 - or represents.
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p -value? In 1 – 2 complete sentences, explain what the p-value means for this problem.
g. Use the previous information to sketch a picture of this situation. CLEARLY label and scale the horizontal axis and shade the region(s) corresponding to the p -value.

Figure 14.3. 

Figure (wave.PNG)

h. Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
i. Alpha:
ii. Decision:
iii. Reason for decision:
iv. Conclusion:
i. In complete sentences, explain how you determined which distribution to use.

Solution Sheet: The Chi-Square Distribution*

Class Time:

Name:

a. H o : _______
b. H a :
c. What are the degrees of freedom?
d. State the distribution to use for the test.
e. What is the test statistic?
f. What is the p -value? In 1 – 2 complete sentences, explain what the p -value means for this problem.
g. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p -value.

Figure 14.4. 

Figure (wave2.PNG)

h. Indicate the correct decision (“reject” or “do not reject” the null hypothesis) and write appropriate conclusions, using complete sentences.
i. Alpha:
ii. Decision:
iii. Reason for decision:
iv. Conclusion:

Solution Sheet: F Distribution and ANOVA*

Class Time:

Name:

a. H o :
b. H a :
c. =
d. =
e. State the distribution to use for the test.
f. What is the test statistic?
g. What is the p -value? In 1 – 2 complete sentences, explain what the p -value means for this problem.
h. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p -value.

Figure 14.5. 

Figure (line1.PNG)

i. Indicate the correct decision (“reject” or “do not reject” the null hypothesis) and write appropriate conclusions, using complete sentences.
i. Alpha:
ii. Decision:
iii. Reason for decision:
iv. Conclusion:

14.6. English Phrases Written Mathematically*

English Phrases Written Mathematically

Table 14.16.
When the English says:Interpret this as:
  
X is at least 4. X ≥ 4
X The minimum is 4. X ≥ 4
X is no less than 4. X ≥ 4
X is greater than or equal to 4. X ≥ 4
  
X is at most 4. X ≤ 4
X The maximum is 4. X ≤ 4
X is no more than 4. X ≤ 4
X is less than or equal to 4. X ≤ 4
X does not exceed 4. X ≤ 4
  
X is greater than 4. X > 4
X There are more than 4. X > 4
X exceeds 4. X > 4
  
X is less than 4. X < 4
X There are fewer than 4. X < 4
  
X is 4. X = 4
X is equal to 4. X = 4
X is the same as 4. X = 4
  
X is not 4. X ≠ 4
X is not equal to 4. X ≠ 4
X is not the same as 4. X ≠ 4
X is different than 4. X ≠ 4
  

14.7. Symbols and their Meanings*

Table 14.17. Symbols and their Meanings
Chapter (1st used)SymbolSpokenMeaning
    
Sampling and Data The square root ofsame
Sampling and Data π Pi3.14159… (a specific number)
Descriptive Statistics Q1 Quartile onethe first quartile
Descriptive Statistics Q2 Quartile twothe second quartile
Descriptive Statistics Q3 Quartile threethe third quartile
Descriptive Statistics IQR inter-quartile rangeQ3-Q1=IQR
Descriptive Statistics x-barsample mean
Descriptive Statistics μ mupopulation mean
Descriptive Statistics s s x sx ssample standard deviation
Descriptive Statistics s 2 s x 2 s-squaredsample variance
Descriptive Statistics σ σ x σx sigmapopulation standard deviation
Descriptive Statistics σ 2 σ x 2 sigma-squaredpopulation variance
Descriptive Statistics Σ capital sigmasum
Probability Topics { } bracketsset notation
Probability Topics S Ssample space
Probability Topics A Event Aevent A
Probability Topics P ( A ) probability of Aprobability of A occurring
Probability Topics P ( AB ) probability of A given Bprob. of A occurring given B has occurred
Probability Topics P ( AorB ) prob. of A or Bprob. of A or B or both occurring
Probability Topics P ( AandB ) prob. of A and Bprob. of both A and B occurring (same time)
Probability Topics A’ A-prime, complement of Acomplement of A, not A
Probability Topics P ( A’ ) prob. of complement of A same
Probability Topics G 1 green on first picksame
Probability Topics prob. of green on first pick same
Discrete Random Variables PDF prob. distribution function same
Discrete Random Variables X Xthe random variable X
Discrete Random Variables X~ the distribution of X same
Discrete Random Variables B binomial distributionsame
Discrete Random Variables G geometric distributionsame
Discrete Random Variables H hypergeometric dist.same
Discrete Random Variables P Poisson dist.same
Discrete Random Variables λ Lambdaaverage of Poisson distribution
Discrete Random Variables greater than or equal tosame
Discrete Random Variables less than or equal tosame
Discrete Random Variables = equal to same
Discrete Random Variables not equal tosame
Continuous Random Variables f ( x ) f of xfunction of x
Continuous Random Variables pdf prob. density function same
Continuous Random Variables U uniform distributionsame
Continuous Random Variables Exp exponential distributionsame
Continuous Random Variables k kcritical value
Continuous Random Variables f ( x ) = f of x equalssame
Continuous Random Variables m mdecay rate (for exp. dist.)
The Normal Distribution N normal distributionsame
The Normal Distribution z z-scoresame
The Normal Distribution Z standard normal dist.same
The Central Limit Theorem Central Limit Theoremsame
The Central Limit Theorem X-barthe random variable X-bar
The Central Limit Theorem μ x mean of Xthe average of X
The Central Limit Theorem mean of X-barthe average of X-bar
The Central Limit Theorem σ x standard deviation of Xsame
The Central Limit Theorem standard deviation of X-bar same
The Central Limit Theorem Σ X sum of Xsame
The Central Limit Theorem Σ x sum of xsame
Confidence Intervals confidence levelsame
Confidence Intervals confidence intervalsame
Confidence Intervals error bound for a meansame
Confidence Intervals error bound for a proportion same
Confidence Intervals t student-t distributionsame
Confidence Intervals degrees of freedomsame
Confidence Intervals student-t with a/2 area in right tail same
Confidence Intervals p’ p-prime; p-hatsample proportion of success
Confidence Intervals q’ q-prime; q-hatsample proportion of failure
Hypothesis Testing H 0 H-naught, H-sub 0null hypothesis
Hypothesis Testing H a H-a, H-sub aalternate hypothesis
Hypothesis Testing H 1 H-1, H-sub 1alternate hypothesis
Hypothesis Testing α alphaprobability of Type I error
Hypothesis Testing β betaprobability of Type II error
Hypothesis Testing X1-bar minus X2-bardifference in sample means
  μ 1 μ 2 mu-1 minus mu-2difference in population means
  P 1 P 2 P1-prime minus P2-primedifference in sample proportions
  p 1 p 2 p1 minus p2difference in population proportions
Chi-Square Distribution Χ 2 Ky-squareChi-square
  O ObservedObserved frequency
  E ExpectedExpected frequency
Linear Regression and Correlation y = a + bx y equals a plus b-xequation of a line
  y-hatestimated value of y
  r correlation coefficientsame
  ε errorsame
  SSE Sum of Squared Errorssame
  1.9 s 1.9 times scut-off value for outliers
F-Distribution and ANOVA F F-ratioF ratio

14.8. Formulas*

Formula 14.1. Factorial

n ! = n ( n − 1 ) ( n − 2 ) . . . ( 1 )

0 ! = 1


Formula 14.2. Combinations


Formula 14.3. Binomial Distribution

X ~ B ( n , p )

, for x = 0 , 1 , 2 , . . . , n


Formula 14.4. Geometric Distribution

X ~ G ( p )

, for x = 1 , 2 , 3 , . . .


Formula 14.5. Hypergeometric Distribution

X~H ( r , b , n )


Formula 14.6. Poisson Distribution

X~P ( μ )


Formula 14.7. Uniform Distribution

X ~ U ( a , b )

, a < x < b


Formula 14.8. Exponential Distribution

X ~ Exp ( m )

, m > 0 , x ≥ 0


Formula 14.9. Normal Distribution

,


Formula 14.10. Gamma Function

z > 0

Γ ( m + 1 ) = m ! for m , a nonnegative integer

otherwise: Γ ( a + 1 ) = aΓ ( a )


Formula 14.11. Student-t Distribution

X ~ t df

Z ~ N ( 0,1 ) , Y ~ Χ df 2 , n = degrees of freedom


Formula 14.12. Chi-Square Distribution

X ~ Χ df 2

, x > 0 , n = positive integer and degrees of freedom


Formula 14.13. F Distribution

X ~ F df ( n ) , df ( d )

df ( n ) = degrees of freedom for the numerator

df ( d ) = degrees of freedom for the denominator

, Y , W are chi-square


14.9. Notes for the TI-83, 83+, 84 Calculator*

Quick Tips

Legend

  • Legend represents a button press

  • [ ] represents yellow command or green letter behind a key

  • < > represents items on the screen

To adjust the contrast

Press To adjust the contrast, then hold To adjust the contrast to increase the contrast or To adjust the contrast to decrease the contrast.

To capitalize letters and words

Press To capitalize letters and words to get one capital letter, or press To capitalize letters and words, then To capitalize letters and words to set all button presses to capital letters. You can return to the top-level button values by pressing To capitalize letters and words again.

To correct a mistake

If you hit a wrong button, just hit To correct a mistake and start again.

To write in scientific notation

Numbers in scientific notation are expressed on the TI-83, 83+, and 84 using E notation, such that…

  • 4.321 E 4 = 4.321 × 104

  • 4.321 E -4 = 4.321 × 10 – 4

To transfer programs or equations from one calculator to another:

Both calculators: Insert your respective end of the link cable cable and press To transfer programs or equations from one calculator to another:, then [LINK].

Calculator receiving information:

  1. Use the arrows to navigate to and select <RECEIVE>

  2. Press Calculator receiving information:

Calculator sending information:

  1. Press appropriate number or letter.

  2. Use up and down arrows to access the appropriate item.

  3. Press Calculator sending information: to select item to transfer.

  4. Press right arrow to navigate to and select <TRANSMIT>.

  5. Press Calculator sending information:

Note

ERROR 35 LINK generally means that the cables have not been inserted far enough.

Both calculators: Insert your respective end of the link cable cable Both calculators: press Calculator sending information:, then [QUIT] To exit when done.

Manipulating One-Variable Statistics

Note

These directions are for entering data with the built-in statistical program.

Table 14.18. Sample Data
We are manipulating 1-variable statistics.
DataFrequency
-210
-13
04
15
38

To begin:

  1. Turn on the calculator.

    To begin:

  2. Access statistics mode.

    To begin:

  3. Select <4:ClrList> to clear data from lists, if desired.

    To begin: , To begin:

  4. Enter list [L1] to be cleared.

    To begin: , [L1] , To begin:

  5. Display last instruction.

    To begin: , [ENTRY]

  6. Continue clearing remaining lists in the same fashion, if desired.

    To begin: , To begin: , [L2] , To begin:

  7. Access statistics mode.

    To begin:

  8. Select <1:Edit . . .>

    To begin:

  9. Enter data. Data values go into [L1]. (You may need to arrow over to [L1])

    • Type in a data value and enter it. (For negative numbers, use the negate (-) key at the bottom of the keypad)

      To begin: , To begin: , To begin:

    • Continue in the same manner until all data values are entered.

  10. In [L2], enter the frequencies for each data value in [L1].

    • Type in a frequency and enter it. (If a data value appears only once, the frequency is “1”)

      To begin: , To begin:

    • Continue in the same manner until all data values are entered.

  11. Access statistics mode.

    To begin:

  12. Navigate to <CALC>

  13. Access <1:1-var Stats>

    To begin:

  14. Indicate that the data is in [L1]

    To begin: , [L1] , To begin:

  15. …and indicate that the frequencies are in [L2].

    To begin: , [L2] , To begin:

  16. The statistics should be displayed. You may arrow down to get remaining statistics. Repeat as necessary.

Drawing Histograms

Note

We will assume that the data is already entered

We will construct 2 histograms with the built-in STATPLOT application. The first way will use the default ZOOM. The second way will involve customizing a new graph.

  1. Access graphing mode.

    Drawing Histograms , [STAT PLOT]

  2. Select <1:plot 1> To access plotting - first graph.

    Drawing Histograms

  3. Use the arrows navigate go to <ON> to turn on Plot 1.

    <ON> , Drawing Histograms

  4. Use the arrows to go to the histogram picture and select the histogram.

    Drawing Histograms

  5. Use the arrows to navigate to <Xlist>

  6. If “L1” is not selected, select it.

    Drawing Histograms , [L1] , Drawing Histograms

  7. Use the arrows to navigate to <Freq>.

  8. Assign the frequencies to [L2].

    Drawing Histograms , [L2] , Drawing Histograms

  9. Go back to access other graphs.

    Drawing Histograms , [STAT PLOT]

  10. Use the arrows to turn off the remaining plots.

  11. Be sure to deselect or clear all equations before graphing.

To deselect equations:

  1. Access the list of equations.

    To deselect equations:

  2. Select each equal sign (=).

    To deselect equations: To deselect equations: To deselect equations:

  3. Continue, until all equations are deselected.

To clear equations:

  1. Access the list of equations.

    To clear equations:

  2. Use the arrow keys to navigate to the right of each equal sign (=) and clear them.

    To clear equations: To clear equations: To clear equations:

  3. Repeat until all equations are deleted.

To draw default histogram:

  1. Access the ZOOM menu.

    To draw default histogram:

  2. Select <9:ZoomStat>

    To draw default histogram:

  3. The histogram will show with a window automatically set.

To draw custom histogram:

  1. Access

    To draw custom histogram:

    to set the graph parameters.

    • X min = – 2.5

    • X max = 3.5

    • X scl = 1 (width of bars)

    • Y min = 0

    • Y max = 10

    • Y scl = 1 (spacing of tick marks on y-axis)

    • X res = 1

  2. Access

    To draw custom histogram:

    to see the histogram.

To draw box plots:

  1. Access graphing mode.

    To draw box plots: , [STAT PLOT]

  2. Select <1:Plot 1> to access the first graph.

    To draw box plots:

  3. Use the arrows to select <ON> and turn on Plot 1.

    To draw box plots:

  4. Use the arrows to select the box plot picture and enable it.

    To draw box plots:

  5. Use the arrows to navigate to <Xlist>

  6. If “L1” is not selected, select it.

    To draw box plots: , [L1] , To draw box plots:

  7. Use the arrows to navigate to <Freq>.

  8. Indicate that the frequencies are in [L2].

    To draw box plots: , [L2] , To draw box plots:

  9. Go back to access other graphs.

    To draw box plots: , [STAT PLOT]

  10. Be sure to deselect or clear all equations before graphing using the method mentioned above.

  11. View the box plot.

    To draw box plots: , [STAT PLOT]

Linear Regression

Sample Data

The following data is real. The percent of declared ethnic minority students at De Anza College for selected years from 1970 - 1995 was:

Table 14.19. The independent variable is “Year,” while the independent variable is “Student Ethnic Minority Percent.”
YearStudent Ethnic Minority Percentage
197014.13
197312.27
197614.08
197918.16
198227.64
198328.72
198631.86
198933.14
199245.37
199553.1

Figure 14.6. Student Ethnic Minority Percentage

Student Ethnic Minority Percentage
By hand, verify the scatterplot above.

Note

The TI-83 has a built-in linear regression feature, which allows the data to be edited.The x-values will be in [L1]; the y-values in [L2].

To enter data and do linear regression:

  1. ON Turns calculator on

    To enter data and do linear regression:

  2. Before accessing this program, be sure to turn off all plots.

    • Access graphing mode.

      To enter data and do linear regression: , [STAT PLOT]

    • Turn off all plots.

      To enter data and do linear regression: , To enter data and do linear regression:

  3. Round to 3 decimal places. To do so:

    • Access the mode menu.

      To enter data and do linear regression: , [STAT PLOT]

    • Navigate to <Float> and then to the right to <3>.

      To enter data and do linear regression: To enter data and do linear regression:

    • All numbers will be rounded to 3 decimal places until changed.

      To enter data and do linear regression:

  4. Enter statistics mode and clear lists [L1] and [L2], as describe above.

    To enter data and do linear regression: , To enter data and do linear regression:

  5. Enter editing mode to insert values for x and y.

    To enter data and do linear regression: , To enter data and do linear regression:

  6. Enter each value. Press

    To enter data and do linear regression:

    to continue.

To display the correlation coefficient:

  1. Access the catalog.

    To display the correlation coefficient: , [CATALOG]

  2. Arrow down and select <DiagnosticOn>

    To display the correlation coefficient:… , To display the correlation coefficient: , To display the correlation coefficient:

  3. r and r 2 will be displayed during regression calculations.

  4. Access linear regression.

    To display the correlation coefficient: To display the correlation coefficient:

  5. Select the form of y = a + bx

    To display the correlation coefficient: , To display the correlation coefficient:

The display will show:

LinReg

  • y = a + bx

  • a = – 3176.909

  • b = 1.617

  • r 2 = 0.924

  • r = 0.961

This means the Line of Best Fit (Least Squares Line) is:

  • y = – 3176.909 + 1.617x

  • Percent = – 3176.909 + 1.617(year #)

The correlation coefficient r = 0.961

To see the scatter plot:

  1. Access graphing mode.

    To see the scatter plot: , [STAT PLOT]

  2. Select <1:plot 1> To access plotting - first graph.

    To see the scatter plot:

  3. Navigate and select <ON> to turn on Plot 1.

    <ON> To see the scatter plot:

  4. Navigate to the first picture.

  5. Select the scatter plot.

    To see the scatter plot:

  6. Navigate to <Xlist>

  7. If [L1] is not selected, press

    To see the scatter plot:

    , [L1] to select it.

  8. Confirm that the data values are in [L1].

    <ON> To see the scatter plot:

  9. Navigate to <Ylist>

  10. Select that the frequencies are in [L2].

    To see the scatter plot: , [L2] , To see the scatter plot:

  11. Go back to access other graphs.

    To see the scatter plot: , [STAT PLOT]

  12. Use the arrows to turn off the remaining plots.

  13. Access

    To see the scatter plot:

    to set the graph parameters.

    • X min = 1970

    • X max = 2000

    • X scl = 10 (spacing of tick marks on x-axis)

    • Y min = – 0.05

    • Y max = 60

    • Y scl = 10 (spacing of tick marks on y-axis)

    • X res = 1

  14. Be sure to deselect or clear all equations before graphing, using the instructions above.

  15. Press

    To see the scatter plot:

    to see the scatter plot.

To see the regression graph:

  1. Access the equation menu. The regression equation will be put into Y1.

    To see the regression graph:

  2. Access the vars menu and navigate to <5: Statistics>

    To see the regression graph: , To see the regression graph:

  3. Navigate to <EQ>.

  4. <1: RegEQ> contains the regression equation which will be entered in Y1.

    To see the regression graph:

  5. Press

    To see the regression graph:

    . The regression line will be superimposed over scatter plot.

To see the residuals and use them to calculate the critical point for an outlier:

  1. Access the list. RESID will be an item on the menu. Navigate to it.

    To see the residuals and use them to calculate the critical point for an outlier:, [LIST], <RESID>

  2. Confirm twice to view the list of residuals. Use the arrows to select them.

    To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier:

  3. The critical point for an outlier is:  where:

    • n = number of pairs of data

    • SSE = sum of the squared errors

    • ∑(residual2)

  4. Store the residuals in [L3].

    To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , [L3] , To see the residuals and use them to calculate the critical point for an outlier:

  5. Calculate the . Note that n − 2 = 8

    To see the residuals and use them to calculate the critical point for an outlier: , [L3] , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier:

  6. Store this value in [L4].

    To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , [L4] , To see the residuals and use them to calculate the critical point for an outlier:

  7. Calculate the critical value using the equation above.

    To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , [V] , To see the residuals and use them to calculate the critical point for an outlier: , [LIST] To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , [L4] , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier: , To see the residuals and use them to calculate the critical point for an outlier:

  8. Verify that the calculator displays: 7.642669563. This is the critical value.

  9. Compare the absolute value of each residual value in [L3] to 7.64 . If the absolute value is greater than 7.64, then the (x, y) corresponding point is an outlier. In this case, none of the points is an outlier.

To obtain estimates of y for various x-values:

There are various ways to determine estimates for “y”. One way is to substitute values for “x” in the equation. Another way is to use the To obtain estimates of y for various x-values: on the graph of the regression line.

TI-83, 83+, 84 instructions for distributions and tests

Distributions

Access DISTR (for “Distributions”).

For technical assistance, visit the Texas Instruments website at http://www.ti.com and enter your calculator model into the “search” box.

Binomial Distribution

  • binompdf(n,p,x) corresponds to P(X = x)

  • binomcdf(n,p,x) corresponds to P(X ≤ x)

  • To see a list of all probabilities for x: 0, 1, … , n, leave off the “x“ parameter.

Poisson Distribution

  • poissonpdf(λ,x) corresponds to P(X = x)

  • poissoncdf(λ,x) corresponds to P(X ≤ x)

Continuous Distributions (general)

  • – ∞ uses the value -1EE99 for left bound

  • + ∞ uses the value 1EE99 for right bound

Normal Distribution

  • normalpdf(x,μ,σ) yields a probability density function value (only useful to plot the normal curve, in which case “x” is the variable)

  • normalcdf(left bound, right bound, μ,σ) corresponds to P(left bound < X < right bound)

  • normalcdf(left bound, right bound) corresponds to P(left bound < Z < right bound) - standard normal

  • invNorm(p,μ,σ) yields the critical value, k: P(X < k) = p

  • invNorm(p) yields the critical value, k: P(Z < k) = p for the standard normal

Student-t Distribution

  • tpdf(x,df) yields the probability density function value (only useful to plot the student-t curve, in which case “x” is the variable)

  • tcdf(left bound, right bound, df) corresponds to P(left bound < t < right bound)

Chi-square Distribution

  • Χ2pdf(x,df) yields the probability density function value (only useful to plot the chi2 curve, in which case “x” is the variable)

  • Χ2cdf(left bound, right bound, df) corresponds to P(left bound < Χ2 < right bound)

F Distribution

  • Fpdf(x,dfnum,dfdenom) yields the probability density function value (only useful to plot the F curve, in which case “x” is the variable)

  • Fcdf(left bound,right bound,dfnum,dfdenom) corresponds to P(left bound < F < right bound)

Tests and Confidence Intervals

Access STAT and TESTS.

For the Confidence Intervals and Hypothesis Tests, you may enter the data into the appropriate lists and press DATA to have the calculator find the sample means and standard deviations. Or, you may enter the sample means and sample standard deviations directly by pressing STAT once in the appropriate tests.

Confidence Intervals

  • ZInterval is the confidence interval for mean when σ is known

  • TInterval is the confidence interval for mean when σ is unknown; s estimates σ.

  • 1-PropZInt is the confidence interval for proportion

Note

The confidence levels should be given as percents (ex. enter “95” for a 95% confidence level).

Hypothesis Tests

  • Z-Test is the hypothesis test for single mean when σ is known

  • T-Test is the hypothesis test for single mean when σ is unknown; s estimates σ.

  • 2-SampZTest is the hypothesis test for 2 independent means when both σ’s are known

  • 2-SampTTest is the hypothesis test for 2 independent means when both σ’s are unknown

  • 1-PropZTest is the hypothesis test for single proportion.

  • 2-PropZTest is the hypothesis test for 2 proportions.

  • Χ2-Test is the hypothesis test for independence.

  • Χ2GOF-Test is the hypothesis test for goodness-of-fit (TI-84+ only).

  • LinRegTTEST is the hypothesis test for Linear Regression (TI-84+ only).

Note

Input the null hypothesis value in the row below “Inpt.” For a test of a single mean, “μ∅” represents the null hypothesis. For a test of a single proportion, “p∅” represents the null hypothesis. Enter the alternate hypothesis on the bottom row.