11-3. Exercises

1. BASIC

Ex 1. Find $F_{0.01}$ for each of the following degrees of freedom.

1. $df_1=5$ and $df_2=5$

2. $df_1=5$ and $df_2=12$

3. $df_1=12$ and $df_2=20$

Ex 2. Find $F_{0.05}$ for each of the following degrees of freedom.

1. $df_1=6$ and $df_2=6$

2. $df_1=6$ and $df_2=12$

3. $df_1=12$ and $df_2=30$

Ex 3. Find $F_{0.95}$ for each of the following degrees of freedom.

1. $df_1=6$ and $df_2=6$

2. $df_1=6$ and $df_2=12$

3. $df_1=12$ and $df_2=30$

Ex 4. Find $F_{0.05}$ for each of the following degrees of freedom.

1. $df_1=5$ and $df_2=5$

2. $df_1=5$ and $df_2=12$

3. $df_1=12$ and $df_2=20$

Ex 5. For $df_1=7$, $df_2=5$ and $α=0.05$ , find

1. $F_α$

2. $F_{1−α}$

3. $F_{α∕2}$

4. $F_{1−α∕2}$

Ex 6. For $df_1=15$, $df_2=8$ and $α=0.01$ , find

1. $F_α$

2. $F_{1−α}$

3. $F_{α∕2}$

4. $F_{1−α∕2}$

Ex 7. For each of the two samples Sample 1 : { 8, 2, 11, 0, −2,} Sample 2 : {−2, 0, 0, 0, 2, 4, −1}

find

1. the sample size,

2. the sample mean,

3. the sample variance.

Ex 8. For each of the two samples Sample 1 : {0.8, 1.2, 1.1, 0.8, −2.0} Sample 2 : {−2.0, 0.0, 0.7, 0.8, 2.2, 4.1, −1.9}

find

1. the sample size,

2. the sample mean,

3. the sample variance.

Ex 9. Two random samples taken from two normal populations yielded the following information:

Sample	Sample Size	Sample Variance
1	$n_1=16$	$s^2_1=53$
2	$n_2=21$	$s_2^2=32$

1. Find the statistic $F=s^2_1/s^2_2$ .

2. Find the degrees of freedom $df_1$ and $df_2$ .

3. Find $F_{0.05}$ using $df_1$ and $df_2$ computed above.

4. Perform the test the hypotheses $H_0:σ^2_1=σ^2_2$ vs. $H_a: σ^2_1>σ^2_2$ at the 5% level of significance.

Ex 10. Two random samples taken from two normal populations yielded the following information:

Sample	Sample Size	Sample Variance
1	$n_1=11$	$s^2_1=61$
2	$n_2=8$	$s_2^2=44$

1. Find the statistic $F=s^2_1/s^2_2$ .

2. Find the degrees of freedom $df_1$ and $df_2$ .

3. Find $F_{0.05}$ using $df_1$ and $df_2$ computed above.

4. Perform the test the hypotheses $H_0:σ^2_1=σ^2_2$ vs. $H_a: σ^2_1>σ^2_2$ at the 5% level of significance.

Ex 11. Two random samples taken from two normal populations yielded the following information:

Sample	Sample Size	Sample Variance
1	$n_1=10$	$s^2_1=12$
2	$n_2=13$	$s_2^2=23$

1. Find the statistic $F=s^2_1/s^2_2$ .

2. Find the degrees of freedom $df_1$ and $df_2$ .

3. For $α=0.05$ find $F_{1-α}$ using $df_1$ and $df_2$ computed above.

4. Perform the test the hypotheses $H_0:σ^2_1=σ^2_2$ vs. $H_a: σ^2_1<σ^2_2$ at the 5% level of significance.

Ex 12. Two random samples taken from two normal populations yielded the following information:

Sample	Sample Size	Sample Variance
1	$n_1=8$	$s^2_1=102$
2	$n_2=8$	$s_2^2=603$

1. Find the statistic $F=s^2_1/s^2_2$ .

2. Find the degrees of freedom $df_1$ and $df_2$ .

3. For $α=0.05$ find $F_{1-α}$ using $df_1$ and $df_2$ computed above.

4. Perform the test the hypotheses $H_0:σ^2_1=σ^2_2$ vs. $H_a: σ^2_1<σ^2_2$ at the 5% level of significance.

Ex 13. Two random samples taken from two normal populations yielded the following information:

Sample	Sample Size	Sample Variance
1	$n_1=9$	$s^2_1=123$
2	$n_2=31$	$s_2^2=543$

1. Find the statistic $F=s^2_1/s^2_2$ .

2. Find the degrees of freedom $df_1$ and $df_2$ .

3. For $α=0.05$ find $F_{1-α/2}$ and $F_{α/2}$using $df_1$ and $df_2$ computed above.

4. Perform the test the hypotheses $H_0:σ^2_1=σ^2_2$ vs. $H_a: σ^2_1\neσ^2_2$ at the 5% level of significance.

Ex 14. Two random samples taken from two normal populations yielded the following information:

Sample	Sample Size	Sample Variance
1	$n_1=21$	$s^2_1=199$
2	$n_2=21$	$s_2^2=66$

1. Find the statistic $F=s^2_1/s^2_2$ .

2. Find the degrees of freedom $df_1$ and $df_2$ .

3. For $α=0.05$ find $F_{1-α/2}$ and $F_{α/2}$using $df_1$ and $df_2$ computed above.

4. Perform the test the hypotheses $H_0:σ^2_1=σ^2_2$ vs. $H_a: σ^2_1\neσ^2_2$ at the 5% level of significance.

2. APPLICATIONS

Ex 15. Japanese sturgeon is a subspecies of the sturgeon family indigenous to Japan and the Northwest Pacific. In a particular fish hatchery newly hatched baby Japanese sturgeon are kept in tanks for several weeks before being transferred to larger ponds. Dissolved oxygen in tank water is very tightly monitored by an electronic system and rigorously maintained at a target level of 6.5 milligrams per liter (mg/l). The fish hatchery looks to upgrade their water monitoring systems for tighter control of dissolved oxygen. A new system is evaluated against the old one currently being used in terms of the variance in measured dissolved oxygen. Thirty-one water samples from a tank operated with the new system were collected and 16 water samples from a tank operated with the old system were collected, all during the course of a day. The samples yield the following information: New Sample 1 : $n_1=31, s_1^2=0.0121$ Old Sample 2 : $n_2=16, s_1^2=0.0319$

Test, at the 10% level of significance, whether the data provide sufficient evidence to conclude that the new system will provide a tighter control of dissolved oxygen in the tanks.

Ex 16. The risk of investing in a stock is measured by the volatility, or the variance, in changes in the price of that stock. Mutual funds are baskets of stocks and offer generally lower risk to investors. Different mutual funds have different focuses and offer different levels of risk. Hippolyta is deciding between two mutual funds, A and B, with similar expected returns. To make a final decision, she examined the annual returns of the two funds during the last ten years and obtained the following information: Mutual Fund A Sample 1 : $n_1=10, s_1^2=0.012$ Mutual Fund B Sample 2 : $n_2=10, s_1^2=0.005$ Test, at the 5% level of significance, whether the data provide sufficient evidence to conclude that the two mutual funds offer different levels of risk.

Ex 17. It is commonly acknowledged that grading of the writing part of a college entrance examination is subject to inconsistency. Every year a large number of potential graders are put through a rigorous training program before being given grading assignments. In order to gauge whether such a training program really enhances consistency in grading, a statistician conducted an experiment in which a reference essay was given to 61 trained graders and 31 untrained graders. Information on the scores given by these graders is summarized below: Trained Sample 1 : $n_1=61, s_1^2=2.15$

Untrained Sample 2 : $n_2=31, s_1^2=3.91$

Test, at the 5% level of significance, whether the data provide sufficient evidence to conclude that the training program enhances the consistency in essay grading.

Ex 18. A common problem encountered by many classical music radio stations is that their listeners belong to an increasingly narrow band of ages in the population. The new general manager of a classical music radio station believed that a new playlist offered by a professional programming agency would attract listeners from a wider range of ages. The new list was used for a year. Two random samples were taken before and after the new playlist was adopted. Information on the ages of the listeners in the sample are summarized below: Before Sample 1 : $n_1=21, s_1^2=56.25$ After Sample 2 : $n_2=16, s_1^2=76.56$

Test, at the 10% level of significance, whether the data provide sufficient evidence to conclude that the new playlist has expanded the range of listener ages.

Ex 19. A laptop computer maker uses battery packs supplied by two companies, A and B. While both brands have the same average battery life between charges (LBC), the computer maker seems to receive more complaints about shorter LBC than expected for battery packs supplied by company B. The computer maker suspects that this could be caused by higher variance in LBC for Brand B. To check that, ten new battery packs from each brand are selected, installed on the same models of laptops, and the laptops are allowed to run until the battery packs are completely discharged. The following are the observed LBCs in hours. Brand A : 3.2 3.4 2.8 3.0 3.0 3.0 2.8 2.9 3.0 3.0 Brand B : 3.0 3.5 2.9 3.1 2.3 2.0 3.0 2.9 3.0 4.1

Test, at the 5% level of significance, whether the data provide sufficient evidence to conclude that the LBCs of Brand B have a larger variance that those of Brand A.

Ex 20. A manufacturer of a blood-pressure measuring device for home use claims that its device is more consistent than that produced by a leading competitor. During a visit to a medical store a potential buyer tried both devices on himself repeatedly during a short period of time. The following are readings of systolic pressure. Manufacturer 132 134 129 129 130 132 Competitor 129 132 129 138

1. Test, at the 5% level of significance, whether the data provide sufficient evidence to conclude that the manufacturer’s claim is true.

2. Repeat the test at the 10% level of significance. Quote as many computations from part (a) as possible.

3. LARGE DATA SET EXERCISES

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data1A.xls

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data1B.xls

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data7.xls

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data7A.xls

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data7B.xls

Ex 21. Large Data Sets 1A and 1B record SAT scores for 419 male and 581 female students. Test, at the 1% level of significance, whether the data provide sufficient evidence to conclude that the variances of scores of male and female students differ.

Ex 22. Large Data Sets 7, 7A, and 7B record the survival times of 140 laboratory mice with thymic leukemia. Test, at the 10% level of significance, whether the data provide sufficient evidence to conclude that the variances of survival times of male mice and female mice differ.