Exam 11: Inferences for Population Standard Deviations

Use the one-standard-deviation chi-square interval procedure to obtain the specified confidence interval for thepopulation standard deviation Ϭ. Assume that the population has a normal distribution. -The mean replacement time for a random sample of 20 washing machines is $10.8$ years and the standard deviation is $2.7$ years. Construct a $99 \%$ confidence interval for the standard deviation, $\sigma$ , of the replacement times of all washing machines of this type.

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has df = (7, 15). Find the F-value having area 0.025 to its right.

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has df = (24, 12). Determine the two F-values that divide the area under the curve into a middle 0.98 area and two outside 0.01 areas.

Use the two-standard-deviations F-interval procedure to find the required confidence interval. Assume that independentsamples have been randomly selected from the two populations and that the variable under consideration is normallydistributed on both populations. -The manager of a juice bottling factory is considering installing a new juice bottling machine which she hopes will reduce the amount of variation in the volumes of juice dispensed into 8-fluid-ounce Bottles. Random samples of 10 bottles filled by the old machine and 9 bottles filled by the new Machine yielded the following volumes of juice (in fluid ounces). Old machine: $\quad 8.1,8.0,8.0,8.1,8.0,7.9,8.0,7.9,8.2,8.0$ New machine: $\quad 8.0,8.1,8.0,8.1,7.9,8.0,7.9,8.0,8.1$ Construct a $90 \%$ confidence interval for the ratio, $\sigma _ { 1 } / \sigma _ { 2 }$ , where $\sigma _ { 1 }$ is the population standard deviation of the volumes of juice dispensed by the old machine and $\sigma _ { 2 }$ is the population standard deviation of the volumes of juice dispensed by the new machine. (Note: $s _ { 1 } = 0.0919 , s _ { 2 } = 0.0782$ )

Use the one-standard-deviation chi-square interval procedure to obtain the specified confidence interval for thepopulation standard deviation Ϭ. Assume that the population has a normal distribution. -A sociologist develops a test to assess attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct a 95% confidence interval for the standard deviation, Ϭ, of the scores of all subjects.

The sample standard deviations and sample sizes are given for independent simple random samples from twopopulations. Use the two-standard-deviations F-test to conduct the required hypothesis test. - $\mathrm { s } _ { 1 } = 5.85 , \mathrm { n } _ { 1 } = 31 , \mathrm {~s} _ { 2 } = 5.01 , \mathrm { n } _ { 2 } = 25$ ; two-tailed test, $\alpha = 0.10$

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has $\mathrm { df } = ( 12,10 )$ . Find the $\mathrm { F }$ -value having area $0.05$ to its right.

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has df = (30, 3). Find the F-value having area 0.90 to its left.

Use the two-standard-deviations F-interval procedure to find the required confidence interval. Assume that independentsamples have been randomly selected from the two populations and that the variable under consideration is normallydistributed on both populations. -A researcher obtained independent random samples of men from two different towns. She recorded the weights of the men. The results are summarized below: =165.1 =159.5 =29.8 =25.2 Construct a $99 \%$ confidence interval for the ratio, $\sigma _ { 1 } / \sigma _ { 2 }$ , where $\sigma _ { 1 }$ is the population standard deviation of the weights of men from town $\mathrm { A }$ and $\sigma _ { 2 }$ is the population standard deviation of the weights of men from town $B$ .

Decide whether applying one-standard-deviation chi-square procedures to the given data appears reasonable. Explainyour answer. -The masses, in grams, of 19 plants of a certain type are given below. 5 6 7 7 8 9 15 18 18 26 28 31 32 33 42 42 43 45 46

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with $\mathrm { df } = 17$ , determine $\chi _ { 0.10 } ^ { 2 }$ .

A $\chi ^ { 2 }$ -curve with 10 degrees of freedom is more skewed than a $\chi ^ { 2 }$ -curve with 12 degrees of freedom.

Use the two-standard-deviations F-interval procedure to find the required confidence interval. Assume that independentsamples have been randomly selected from the two populations and that the variable under consideration is normallydistributed on both populations. -A researcher is interested in comparing the amount of variation in women's scores on a certain test and the amount of variation in men's scores on the same test. Independent random samples of 11 Men and 13 women yielded the following scores. Men: 72,60,52,87,66,74,95,50,81,70,72 Women: 70,78,62,96,75,68,41,74,80,47,73,94,65 Construct a $95 \%$ confidence interval for the ratio, $\sigma _ { 1 } / \sigma _ { 2 }$ , where $\sigma _ { 1 }$ is the population standard deviation of the scores for men and $\sigma _ { 2 }$ is the population standard deviation of the scores for women. (Note: $s _ { 1 } = 13.754$ and $s _ { 2 } = 15.588$ )

A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$ interval procedure to obtainthe specified confidence interval. - $\mathrm { s } = 9 , \mathrm { n } = 10,90 \%$ confidence interval

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has df = (60, 10). Find the F-value having area 0.025 to its left.

Suppose that you are performing a $\chi ^ { 2 }$ -test for a population standard deviation. The hypotheses are as follows. :\sigma=2 :\sigma<2 Which of the following statements regarding the criterion for rejecting the null hypothesis is true?

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has df $= ( 9,7 )$ . Find $\mathrm { F } _ { 0.01 }$ .

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 6 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.90$ to its left.

Perform the required chi-square hypothesis test. Preliminary data analyses and other information indicate that it isreasonable to assume that the variable under consideration is normally distributed. Use the critical-value approach or theP-value approach as indicated. -In 2000, the standard deviation of the scores of all students taking a particular test was 20.3. In 2005, the standard deviation of the scores of a random sample of 18 students taking the same test was $s = 27.1$ . At the $5 \%$ level of significance, do the data provide sufficient evidence to conclude that the standard deviation, $\sigma$ , of all 2005 scores is different from the 2000 standard deviation of $20.3$ ? Use the P-value approach.