Exam 11: Inferences for Population Standard Deviations

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with $d f = 8$ , determine $\chi _ { 0.99 } ^ { 2 }$ .

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 13 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.025$ 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. -When 16 randomly selected customers enter a single main waiting line at a bank, their waiting times have a standard deviation of $2.38$ minutes. When 25 randomly selected customers enter any one of several waiting lines, their waiting times have a standard deviation of $5.35$ minutes. Construct a $98 \%$ confidence interval for the ratio, $\sigma _ { 1 } / \sigma _ { 2 }$ , where $\sigma _ { 1 }$ is the population standard deviation of the waiting times when a single line is used and $\sigma _ { 2 }$ is the population standard deviation of the waiting times when several lines are used.

A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$ -test to conduct the requiredhypothesis test. - $\mathrm { s } = 2.4 , \mathrm { n } = 17$ $\mathrm { H } _ { 0 } : \sigma = 3.5 , \mathrm { H } _ { \mathrm { a } } : \sigma \neq 3.5 , \alpha = 0.10$

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has $\mathrm { df } = ( 4,15 )$ . Find the F-value having area $0.975$ to its left.

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 4 degrees of freedom, determine the two $\chi ^ { 2 }$ -values that divide the area under the curve into a middle $0.98$ area and two outside $0.01$ areas.

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with $\mathrm { df } = 5$ , determine $\chi _ { 0.995 } ^ { 2 }$

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 17 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.05$ to its right.

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. -When 12 bolts are tested for hardness, their indexes have a standard deviation of 41.7. Test the claim that the standard deviation of the hardness indexes for all such bolts is greater than 30.0. Use a 0.025 level of significance. Use the critical-value approach.

Use the one-standard-deviation chi-square interval procedure to obtain the specified confidence interval for thepopulation standard deviation $\sigma$ . . Assume that the population has a normal distribution. -The daily intakes of milk (in ounces)for ten five-year old children selected at random from one school were: 29.1 27.8 13.0 19.4 22.7 19.1 25.5 14.2 19.2 20.3 Find a $99 \%$ confidence interval for the standard deviation, $\sigma$ , of the daily milk intakes of all five-year olds at this school.

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 systolic blood pressure for a random sample of 28 women aged 18-24 is 114.8 mm Hg and the standard deviation is 12.6 mm Hg. Construct a 90% confidence interval for the standard Deviation Ϭ, of the systolic blood pressures of all women aged 18-24.

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. - $\mathrm { s } _ { 1 } = 19.8 , \mathrm { n } _ { 1 } = 16 , \mathrm {~s} _ { 2 } = 22.2 , \mathrm { n } _ { 2 } = 13$ ; left-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 df = (9, 12). Find the F-value having area 0.005 to its left.

The two $\chi ^ { 2 }$ -values that divide the area under a $\chi ^ { 2 }$ -curve into a middle $0.9$ area and two outside $0.05$ areas are $\chi _ { 0.9 } ^ { 2 }$ and $\chi _ { 0.1 } ^ { 2 }$ .

Use the one-standard-deviation chi-square interval procedure to obtain the specified confidence interval for the population standard deviation $\sigma$ . Assume that the population has a normal distribution. -The weights of 22 randomly selected eggs have a mean, $\bar { x }$ , of $1.56 \mathrm { oz }$ and a standard deviation, s, of $0.49$ oz. Determine a $95 \%$ confidence interval for the standard deviation, $\sigma$ , of the weights of all such eggs.

A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$ -test to conduct the required hypothesis test. - $\mathrm { s } = 12 , \mathrm { n } = 13$ $\mathrm { H } _ { 0 } : \sigma = 7 , \mathrm { H } _ { \mathrm { a } } : \sigma > 7 , \alpha = 0.05$

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. - $\mathrm { s } _ { 1 } = 5.17 , \mathrm { n } _ { 1 } = 25 , \mathrm {~s} _ { 2 } = 2.35 , \mathrm { n } _ { 2 } = 17$ ; right-tailed test, $\alpha = 0.01$

Provide an appropriate response. -The $\chi ^ { 2 }$ test for one population standard deviation is robust to moderate violations of the normality assumption.

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 weights of 14 men selected at random from one town have a mean, $\overline { x _ { t } }$ , of 159.3 lb and a standard deviation, s, of 13.5 lb. Determine a 90% confidence interval for the standard deviation, Ϭ, of the Weights of all men from this town.

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. - $\mathrm { s } _ { 1 } = 0.0782 , \mathrm { n } _ { 1 } = 9$ , $\mathrm { s } _ { 2 } = 0.2300 , \mathrm { n } _ { 2 } = 10$ ; left-tailed test, $\alpha = 0.05$