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Homework answers / question archive / Assignment: Section 9
Assignment: Section 9.5 Homework
1) Answer the following questions on the F test statistic. Complete parts (a) through (c) below.
a. If s21 represents the larger of two sample variances, can the F test statistic ever be less than 1?
A. No, because the ratio s21/s22 will always be greater than 1.
B. Yes, because the ratio s22/s12 will always be less than 1.
C. No, because the ratio s22/s21 will always be greater than 1.
D. Yes, because the ratio s21/s22 will always be less than 1.
b. Can the F test statistic ever be a negative number?
A. No, because sample variances are always negative, and the result of dividing the squares of two negative numbers is never negative.
B. No, because sample variances cannot be negative, and the result of dividing the squares of two nonnegative numbers is never negative.
C. Yes, because the numerator and denominator used to calculate the F test statistic can both take on negative and positive values.
D. Yes, because s21 can take on values from - 1 to 1 inclusive.
c. If testing the claim that σ12 ≠ s22, what do we know about the two samples if the test statistic is F 1?
A. The two samples come from populations with the same standard deviation (or variance).
B. The two samples have the same standard deviation (or variance).
C. The two samples have fairly different standard deviations (or variances).
D. The two samples have radically different standard deviations (or variances).
YOU ANSWERED: C.
2) Statistical software was used to evaluate two samples that may have the same standard deviation. Use a 0.025 significance level to test the claim that the standard deviations are the same.
F » 7.4976 p » 0.0724
S1 » 0.007365 s2 » 0.003182
X1 » 0.8456 x2 » 0.7835
Can it be said that the two standard deviations are different?
A. No, because the null hypothesis is rejected.
B. No, because the null hypothesis is not rejected.
C. Yes, because the null hypothesis is rejected.
D. Yes, because the null hypothesis is not rejected.
YOU ANSWERED: D.
3) Words were displayed on a computer screen with background colors of red and blue. Results from scores on a test of word recall are given below. Use a 0.05 significance level to test the claim that the samples are from populations with the same standard deviation. Assume that both samples are independent simple random samples from populations having normal distributions. Does the background color appear to have an effect on the variation of word recall scores?
|
|
n |
x |
s |
|
Red Background |
35 |
15.36 |
5.98 |
|
Blue Background |
36 |
12.42 |
5.42 |
What are the null and alternative hypotheses?
A. H0: σ21 = σ22
H1: σ12 ³ σ22
B. H0: σ21 = σ22
H1: σ21 ¹ σ22
C. H0: σ21 ¹ σ22
H1: σ21 = σ22
D. H0: σ21 = σ22
H1: σ21 < σ22
Identify the test statistic.
F = ________________ (Round to two decimal places as needed.)
Use technology to identify the P-value.
The P-value is ___________. (Round to three decimal places as needed.)
What is the conclusion for this hypothesis test?
A. Fail to reject H0. There is insufficient evidence to warrant rejection of the claim that the samples are from populations with the same standard deviation.
B. Reject H0. There is sufficient evidence to warrant rejection of the claim that the samples are from populations with the same standard deviation.
C. Fail to reject H0. There is sufficient evidence to warrant rejection of the claim that the samples are from populations with the same standard deviation.
D. Reject H0. There is sufficient evidence to warrant rejection of the claim that the samples are from populations with the same standard deviation.
Does the background color appear to have an effect on the variation of word recall scores?
A. No, the variation of word recall scores appears to be the same when the color is changed.
B. Yes, the variation of word recall scores appears to be the same when the color is changed.
C. Yes, the variation of word recall scores appears to be different when the color is changed.
D. No, the variation of word recall scores appears to be different when the color is changed.
YOU ANSWERED: C.
4) The accompanying table gives results from a study of words spoken in a day by men and women. Assume that both samples are independent simple random samples from populations having normal distributions. Use a 0.05 significance level to test the claim that the numbers of words spoken in a day by men vary more than the numbers of words spoken in a day by women.
|
|
n |
x |
s |
|
Men |
185 |
15,668.7 |
8,632.8 |
|
Women |
211 |
16,215.7 |
7,301.2 |
What are the null and alternative hypotheses?
A. H0: σ21 ¹ σ22
H1: σ12 = σ22
B. H0: σ21 = σ22
H1: σ21 > σ22
C. H0: σ21 = σ22
H1: σ21 < σ22
D. H0: σ21 = σ22
H1: σ21 ¹ σ22
Identify the test statistic.
F = 1.40 (Round to two decimal places as needed.)
Use technology to identify the P-value.
The P-value is _____. (Round to three decimal places as needed.)
What is the conclusion for this hypothesis test?
A. Fail to reject H0. There is sufficient evidence to support the claim that the numbers of words spoken in a day by men vary more than the numbers of words spoken in a day by women.
B. Fail to reject H0. There is sufficient evidence to support the claim that the numbers of words spoken in a day by men vary more than the numbers of words spoken in a day by women.
C. Reject H0. There is sufficient evidence to support the claim that the numbers of words spoken in a day by men vary more than the numbers of words spoken in a day by women.
D. Reject H0. There is sufficient evidence to support the claim that the numbers of words spoken in a day by men vary more than the numbers of words spoken in a day by women.
YOU ANSWERED: A.
3.679
5) Researchers conducted an experiment to test the effects of alcohol. The errors were recorded in a test of visual and motor skills for a treatment group of 24 people who drank ethanol and another group of 22 people given a placebo. The errors for the treatment group have a standard deviation 2.18 of, and the errors for the placebo group have a standard deviation of 1.38. Use a 0.05 significance level to test the claim that the treatment group has errors that vary more than the errors of the placebo group. Assume that both samples are independent simple random samples from populations having normal distributions.
What are the null and alternative hypotheses?
A. H0: σ21 ¹ σ22
H1: σ12 = σ22
B. H0: σ21 = σ22
H1: σ21 > σ22
C. H0: σ21 = σ22
H1: σ21 ¹ σ22
D. H0: σ21 = σ22
H1: σ21 < σ22
Identify the test statistic.
F = ____________ (Round to two decimal places as needed.)
Use technology to identify the P-value.
The P-value is __________. (Round to three decimal places as needed.)
What is the conclusion for this hypothesis test?
A. Fail to reject H0. There is sufficient evidence to support the claim that the treatment group has errors that vary more than the errors of the placebo group.
B. Reject H0. There is insufficient evidence to support the claim that the treatment group has errors that vary more than the errors of the placebo group.
C. Fail to reject H0. There is insufficient evidence to support the claim that the treatment group has errors that vary more than the errors of the placebo group.
D. Reject H0. There is sufficient evidence to support the claim that the treatment group has errors that vary more than the errors of the placebo group.
6) A study was conducted to determine whether magnets were effective in treating pain. The values represent measurements of pain using the visual analog scale. Assume that both samples are independent simple random samples from populations having normal distributions. Use a 0.05 significance level to test the claim that those given a sham treatment have pain reductions that vary more than the pain reductions for those treated with magnets.
|
|
n |
x |
s |
|
Sham |
20 |
0.42 |
1.46 |
|
Magnet |
20 |
0.46 |
0.97 |
What are the null and alternative hypotheses?
A. H0: σ21 = σ22
H1: σ12 ¹ σ22
B. H0: σ21 ¹ σ22
H1: σ21 = σ22
C. H0: σ21 = σ22
H1: σ21 < σ22
D. H0: σ21 = σ22
H1: σ21 > σ22
Identify the test statistic.
F = ____________(Round to two decimal places as needed.)
Use technology to identify the P-value.
The P-value is _________. (Round to three decimal places as needed.)
What is the conclusion for this hypothesis test?
A. Fail to reject H0. There is sufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.
B. Reject H0. There is insufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.
C. Fail to reject H0. There is insufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.
D. Reject H0. There is sufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.
7) Researchers measured skulls from different time periods in an attempt to determine whether interbreeding of cultures occurred. Results are given below. Assume that both samples are independent simple random samples from populations having normal distributions. Use a 0.05 significance level to test the claim that the variation of maximal skull breadths in 4000 B.C. is the same as the variation in A.D. 150.
|
|
n |
x |
s |
|
4000 B.C. |
30 |
131.25 mm |
5.12 mm |
|
A.D. |
30 |
136.94 mm |
5.32 mm |
What are the null and alternative hypotheses?
A. H0: σ21 ¹ σ22
H1: σ12 = σ22
B. H0: σ21 = σ22
H1: σ21 ¹ σ22
C. H0: σ21 = σ22
H1: σ21 < σ22
D. H0: σ21 = σ22
H1: σ21 > σ22
Identify the test statistic.
F = __________(Round to two decimal places as needed.)
Use technology to identify the P-value.
The P-value is ___________. (Round to three decimal places as needed.)
What is the conclusion for this hypothesis test?
A. Fail to reject H0. There is insufficient evidence to warrant rejection of the claim that the variation of maximal skull breadths in 4000 B.C. is the same as the variation in A.D. 150.
B. Reject H0. There is sufficient evidence to warrant rejection of the claim that the variation of maximal skull breadths in 4000 B.C. is the same as the variation in A.D. 150.
C. Reject H0. There is insufficient evidence to warrant rejection of the claim that the variation of maximal skull breadths in 4000 B.C. is the same as the variation in A.D. 150.
D. Fail to reject H0. There is sufficient evidence to warrant rejection of the claim that the variation of maximal skull breadths in 4000 B.C. is the same as the variation in A.D. 150.
8) Listed below are amounts of strontium-90 (in millibecquerels or mBq per gram of calcium) in a simple random sample of baby teeth obtained from residents of state A and state B. Use a 0.05 significance level to test the claim that amounts of Strontium-90 from state A residents vary more than amounts from state B residents. Assume that both samples are independent simple random samples from populations having normal distributions.
Click the icon to view the sample data.
What are the null and alternative hypotheses?
A. H0: σ21 ¹ σ22
H1: σ12 = σ22
B. H0: σ21 = σ22
H1: σ21 ¹ σ22
C. H0: σ21 = σ22
H1: σ21 > σ22
D. H0: σ21 = σ22
H1: σ21 < σ22
Identify the test statistic.
F = ______________________ (Round to two decimal places as needed.)
Use technology to identify the P-value.
The P-value is ___________. (Round to three decimal places as needed.)
What is the conclusion for this hypothesis test?
A. Fail to reject H0. There is insufficient evidence to support the claim that amounts of Strontium-90 from state A residents vary more than amounts from state B residents.
B. Reject H0. There is insufficient evidence to support the claim that amounts of Strontium-90 from state A residents vary more than amounts from state B residents.
C. Fail to reject H0. There is sufficient evidence to support the claim that amounts of Strontium-90 from state A residents vary more than amounts from state B residents.
D. Reject H0. There is sufficient evidence to support the claim that amounts of Strontium-90 from state A residents vary more than amounts from state B residents.
1: Sample data
|
State A: |
151 |
142 |
148 |
160 |
151 |
138 |
152 |
149 |
155 |
141 |
139 |
139 |
|
State B: |
136 |
141 |
142 |
132 |
135 |
129 |
129 |
139 |
140 |
142 |
136 |
142 |
9) Use the sample weights (in kg) of male and female college students measured in April of their freshman year, as listed below. Use a 0.05 significance level to test the claim that near the end of the freshman year, weights of male college students vary more than weights of female college students. Assume that both samples are independent simple random samples from populations having normal distributions.
Click the icon to view the sample data.
What are the null and alternative hypotheses?
A. H0: σ21 = σ22
H1: σ12 ¹ σ22
B. H0: σ21 ¹ σ22
H1: σ21 = σ22
C. H0: σ21 = σ22
H1: σ21 < σ22
D. H0: σ21 = σ22
H1: σ21 > σ22
Identify the test statistic.
F = ________________(Round to two decimal places as needed.)
Use technology to identify the P-value.
The P-value is ________________. (Round to three decimal places as needed.)
What is the conclusion for this hypothesis test?
A. Fail to reject H0. There is insufficient evidence to support the claim that near the end of the freshman year, weights of male college students vary more than weights of female college students.
B. Fail to reject H0. There is insufficient evidence to support the claim that near the end of the freshman year, weights of male college students vary more than weights of female college students.
C. Reject H0. There is sufficient evidence to support the claim that near the end of the freshman year, weights of male college students vary more than weights of female college students.
D. Reject H0. There is insufficient evidence to support the claim that near the end of the freshman year, weights of male college students vary more than weights of female college students.
2: Sample data
Sex Weight Sex Weight Sex Weight Sex Weight
M 81 F 56 M 56 M 77
M 86 F 50 M 75 F 60
M 69 M 68 M 79 M 71
M 93 F 69 F 66 M 95
F 68 M 88 F 54 F 69
M 55 M 82 F 64 M 82
F 60 M 92 F 68 M 75
F 53 F 53 F 56
F 68 M 71 F 65
F 56 F 64 F 58
F 47 F 60 M 77
M 69 M 69 M 78
M 66 M 69 M 68
F 55 F 56 M 68
F 57 M 97 F 56
F 60 M 66 F 68
F 52 F 59 F 64
M 84 F 65 M 68
F 58 F 56 M 71
M 67 F 64 F 57
10) The data shows the weights of samples of two different sodas. Test the claim that the weights of the two sodas have the same standard deviation using a "count five" test
|
x |
0.7824 |
0.7837 |
0.7976 |
0.7998 |
0.8058 |
0.8169 |
0.8118 |
0.7858 |
0.8024 |
0.7874 |
|
y |
0.8007 |
0.8003 |
0.8162 |
0.8439 |
0.8191 |
0.7863 |
0.8083 |
0.8203 |
0.8244 |
0.8268 |
a. For the first sample, find the mean absolute deviation (MAD) of each value. Do the same for the second sample.
Find the MAD of each x value. Recall that the MAD of the sample value is |x – x|.
x 0.7824 0.7837 0.7976 0.7998 0.8058 0.8769 0.8118
MAD ______ ______ _____ ____ _____ _____ ________
(Round to four decimal places as needed.)
Find the MAD of each y value.
y 0.8007 0.8003 0.8162 0.8439 0.8191 0.7863 0.8083
MAD ______ ______ _____ ______ ______ _____ _____
(Round to four decimal places as needed.)
b. Let c1 be the number of MAD values in the first sample that are greater than the largest MAD value in the second sample. Also, let c2 be the number of MAD values in the second sample that are greater than the largest MAD value in the first sample.
C1 =
C2 =
c. If the sample sizes are equal, use a critical value of 5. If not, use the formula below to find the critical value.
log (α / 2)/log(n1/n1+n2)
The critical value is ____________.
(Type an integer or decimal rounded to one decimal place as needed.)
d. If c1 is greater than or equal to the critical value, then conclude that s21 > s22. If c2 is greater than or equal to the critical value, then conclude that s22 > s21. Otherwise, fail to reject the null hypothesis that s21 > s22. Which conclusion can be drawn? Choose the correct answer below.
A. reject σ21 = s22 in favor of s22 > s21
B. reject σ21 = in favor of s21 > s22
C. fail to reject σ21 = s22
11) The accompanying data table includes weights (in grams) of a simple random sample of 40 quarters made before 1964 and weights of 40 quarters made after 1964. When designing coin vending machines, the standard deviations of pre-1964 quarters and post-1964 quarters must be considered. Use the Levene-Brown-Forsythe test and a 0.05 significance level to test the claim that the weights of pre-1964 quarters and the weights of post-1964 quarters are from populations with the same standard deviation.
Click the icon to view the data table of pre- and post-1964 quarter weights (grams).
Let the weights of pre-1964 quarters represent population 1. Let the weights of post-1964 quarters represent population 2.
What are the null and alternative hypotheses?
A. H0: σ21 = σ22
H1: σ12 ¹ σ22
B. H0: σ21 = σ22
H1: σ21 < σ22
C. H0: σ21 = σ22
H1: σ21 > σ22
Identify the test statistic.
t = _______________
(Round to three decimal places as needed.)
Identify the P-value.
P-value = _________________
(Round to four decimal places as needed.)
What is the conclusion for this test?
(1) ________________ H0. There (2) ________________ sufficient evidence to warrant rejection of the claim that the variation among weights of quarters made after 1964 is the same as the variation among weights of quarters made before 1964.
3: Data Table
Weight of Pre-1964 Quarters (grams) Weight of Post-1964 Quarters (grams)
6.2738 6.2682 5.7009 5.5571
6.2372 6.2687 5.7444 5.5814
6.1484 6.1967 5.7055 5.6855
6.0025 6.2465 5.5963 5.6274
6.1246 6.3172 5.7242 5.6157
6.2151 6.1487 5.6114 5.6668
6.2866 6.0829 5.6160 5.7198
6.0760 6.1423 5.5999 5.6694
6.1426 6.1970 5.7790 5.5454
6.3415 6.2441 5.6841 5.6646
6.1309 6.3669 5.6234 5.5636
6.2412 6.0775 5.5928 5.6485
6.1442 6.1095 5.6486 5.6703
6.1073 6.1787 5.6661 5.6848
6.1181 6.2130 5.5361 5.5609
6.1352 6.1947 5.5491 5.7344
6.2821 6.1940 5.7239 5.6449
6.2647 6.0257 5.6555 5.5804
6.2908 6.1699 5.6063 5.6051
6.1691 6.3288 5.5712 5.6037
1) Reject
Fail to reject
2) is not
Is
12) Which of the following is NOT true of the F test?
Choose the correct answer below.
A. This test is very resistant to departures from normal distributions.
B. This test uses the F distribution.
C. This test requires that both populations have normal distributions.
D. This test compares two population variances or standard deviations.
13) Which of the following is NOT a requirement for testing a claim about two population standard deviations or variances?
Choose the correct answer below.
A. This test requires that both populations have normal distributions.
B. One of the populations is normally distributed.
C. The two samples are simple random samples.
D. The populations are independent.
14) Which of the following is NOT a property of the F distribution?
Choose the correct answer below.
A. The exact shape of the F distribution depends on the two different degrees of freedom.
B. The F distribution is bell-shaped.
C. Values of the F distribution cannot be negative.
D. The F distribution is not symmetric.
15) Which of the following is NOT true of the F test?
Choose the correct answer below.
A. The count five method is a relatively simple alternative to the F test, and it does not require normally distributed populations.
B. Small values of F are evidence against σ21 = s22.
C. All one-tailed F tests will be right-tailed.
YOU ANSWERED: C.
