Exam 11: Inferences for Population Standard Deviations

The F-test for two population standard deviations is robust to moderate violations of the normality assumption.

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. - $\mathrm { s } _ { 1 } = 32.2 , \mathrm { n } _ { 1 } = 61 , \mathrm {~s} _ { 2 } = 26.8 , \mathrm { n } _ { 2 } = 31 \text {; right-tailed test, } \alpha = 0.01$

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has $\mathrm { df } = ( 20,9 )$ . Find $\mathrm { F } _ { 0.05 }$ .

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. -Heights of men aged 25 to 34 have a standard deviation of $2.9$ in. Use a $0.05$ significance level to test whether the heights of women aged 25 to 34 have a different standard deviation. The heights (in inches) of 16 randomly selected women aged 25 to 34 are listed below. Use the critical-value approach. (Note: s = $2.279$ in.) 62.13 65.09 64.18 66.72 63.09 61.15 67.50 64.65 63.80 64.21 60.17 68.28 66.49 62.10 65.73 64.72

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 = (12, 15). Find F0.005 .

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. -A manufacturer uses a new production method to produce steel rods. Under the old production method, the standard deviation of the lengths of the rods was 3.5 cm. The standard deviation of the lengths of a random sample of 17 steel rods produced under the new method was 2.4 cm. At the 0.10 significance level, test whether the standard deviation of the lengths of the new rods is different from 3.5 cm. Use the critical-value approach.

Decide whether applying one-standard-deviation chi-square procedures to the given data appears reasonable. Explainyour answer. -Test scores for a random sample of 12 students were as follows: 22 30 34 37 40 45 49 53 58 65 74 87

Perform the required hypothesis test for two population standard deviations. Assume that independent samples havebeen randomly selected from the two populations and that the variable under consideration is normally distributed onboth populations. Use the critical-value approach. -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 9 bottles filled by the new machine and 10 bottles filled by the old machine yielded the following volumes of juice (in fluid ounces). New machine: $\quad 8.0,8.1,8.0,8.1,7.9,8.0,7.9,8.0,8.1$ Old machine: $\quad 8.2,8.0,7.9,7.8,8.5,8.1,8.1,8.2,7.9,7.9$ Do the data provide sufficient evidence to conclude that there is less variation in the volumes of juice filled by the new machine than in the volumes of juice filled by the old machine? Perform an F-test at the $10 \%$ significance level. (Note: $\mathrm { s } _ { 1 } = 0.0782 , \mathrm {~s} _ { 2 } = 0.2066$ )

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 9 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.975$ to its right.

Provide an appropriate response. -A machine dispenses a liquid drug into bottles in such a way that the standard deviation of the volumes dispensed is 81 microliters. A new machine is tested on a sample of 24 containers and the standard deviation for this sample group is found to be 50 microliters. Why would it be important to perform a hypothesis test to determine whether the standard deviation, Ϭ, of the volumes dispensed by the new machine is less than 81 microliters?

Perform the required hypothesis test for two population standard deviations. Assume that independent samples havebeen randomly selected from the two populations and that the variable under consideration is normally distributed onboth populations. Use the critical-value approach. -When 25 randomly selected customers enter any one of several waiting lines, their waiting times have a standard deviation of 5.21 minutes. When 17 randomly selected customers enter a single main waiting line, their waiting times have a standard deviation of 2.35 minutes. Do the data provide sufficient evidence to conclude that there is more variation in the waiting times when several lines are used? Perform an F-test at the 1% significance level.

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 20 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.99$ to its left.

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 amounts (in ounces) of juice in eight randomly selected juice bottles are: 15.3 15.3 15.8 15.2 15.4 15.3 15.9 15.2 Find a $98 \%$ confidence interval for the standard deviation, $\sigma$ , of the amounts of juice in all such bottles.

A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$ -test to conduct the required hvpothesis test. - $\mathrm { s } = 6 , \mathrm { n } = 10$ $\mathrm { H } _ { 0 } : \sigma = 7 , \mathrm { H } _ { \mathrm { a } } : \sigma \neq 7 , \alpha = 0.05$

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. -A machine dispenses a liquid drug into bottles in such a way that the standard deviation of the contents is 81 milliliters. A new machine is tested on a sample of 24 containers and the standard deviation for this sample group is found to be 26 milliliters. At the 5% level of significance, do the data provide sufficient evidence to conclude that the standard deviation, $\sigma _ {,}$ of the amounts dispensed by the new machine is less than 81 milliliters? Use the critical-value approach.

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 19 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.005$ to its left.

$\mathrm { F } _ { 0.99 }$ for an $\mathrm { F }$ -curve with $\mathrm { df } = ( 25,26 )$ is equal to $1 / \mathrm { F } _ { 0.01 }$ for an $\mathrm { F }$ -curve with $\mathrm { df } = ( 26,25 )$ .

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. -With individual lines at the checkouts, a store manager finds that the standard deviation for the waiting times on Monday mornings is 5.2 minutes. After switching to a single waiting line, he finds that for a random sample of 29 customers, the waiting times have a standard deviation of 4.6 minutes. Use a 0.025 significance level to test whether the standard deviation of the waiting times using a single line is less than 5.2 minutes. Use the critical-value approach.

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 = (3, 5). Find the F-value having area 0.01 to its right.