Exam 11: Inferences for Population Standard Deviations

A hypothesis test for two population standard deviations is to be performed. Independent random samples of sizes $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$ are drawn from the two populations. The variable under consideration is normally distributed on each of the two populations. The hypotheses are: := :< Which of the following would provide evidence against the null hypothesis in favor of the alternative?

A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$ -interval procedure to obtainthe specified confidence interval. -s = 6, n = 18 , 99% confidence interval

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. -The standard deviation of math test scores at one high school is 16.1. The standard deviation of the math scores of a random sample of 22 girls was 12. Use a significance level of 0.01 to test whether the standard deviation, Ϭ, of the girls' test scores is smaller than 16.1. Use the critical-value approach.

For a given $\chi ^ { 2 }$ -curve, rank the following $\chi ^ { 2 }$ -values in ascending order: $\chi _ { 0.01 } ^ { 2 } , \chi _ { 0.9 } ^ { 2 } , \chi _ { 0.1 } ^ { 2 }$ ,

Use the F-table and the reciprocal property of F-curves, if necessary, to find the required F-value(s). -An F-curve has $\mathrm { df } = ( 24,20 )$ . Find $\mathrm { F } _ { 0.10 } { } ^ { \circ }$

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 8 9 15 17 18 18 21 24 25 26 28 28 31 32 33 35 42 42

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: 24 42 52 55 60 65 67 71 72 75 77 79

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 football coach randomly selected ten players from his college and timed how long each player took to perform a certain drill. The times (in minutes)were: 6 14 10 9 8 15 11 10 13 10 Find a $95 \%$ confidence interval for the standard deviation, $\sigma$ , of the times of all football players at this college.

Provide an appropriate response. -When you invest your money in a savings account, the standard deviation of the annual return is small. When you invest your money in the stock market, the standard deviation of your annual return can be high. In which of the situations below would knowledge of the standard deviation be more important? Explain your thinking. A: A 25-year old man will invest money in a mutual fund and plans to leave his money there for a long period. B: A 63-year old man nearing retirement will invest money in a mutual fund. He will only be able to invest for a short period and will then need to withdraw his money.

An F-curve is right-skewed if the numerator degrees of freedom are larger than the denominator degrees of freedom and left-skewed if the denominator degrees of freedom are larger than the numerator degrees of freedom.

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, 10). Determine the two F-values that divide the area under the curve into a middle 0.99 area and two outside 0.005 areas.

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 one town, monthly incomes for men with college degrees are found to have a standard deviation of $650. A random sample of 22 men without college degrees resulted in incomes with a standard deviation of $905. At the 1% level of significance, do the data provide sufficient evidence to conclude that the standard deviation, Ϭ, of incomes of men in that town without college degrees is greater than $650? Use the critical-value approach.

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. -For randomly selected adults, IQ scores are normally distributed with a standard deviation of 15. The scores of 14 randomly selected college students are listed below. Use a $0.10$ significance level to test the claim that the standard deviation of IQ scores of college students is less than 15. Use the P-value approach. (Note: $\mathrm { s } = 9.819$ ) 115 128 107 109 116 124 135 127 115 104 118 126 129 133

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 = (6, 24). Find the F-value having area 0.01 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. -Systolic blood pressure levels for men have a standard deviation of 19.7 mm Hg. A random sample of 31 women resulted in systolic blood pressure levels with a standard deviation of 22.1 mm Hg. Use a 0.05 significance level to test whether the standard deviation of systolic blood pressure levels for women is different from 19.7 mm Hg. Use the critical-value approach.

A hypothesis test for two population standard deviations is to be performed. Independent random samples of sizes $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$ are drawn from the two populations. The variable under consideration is normally distributed on each of the two populations. The test statistic is $\mathrm { s } _ { 1 } ^ { 2 } / \mathrm { s } _ { 2 } ^ { 2 }$ . If the null hypothesis, $\mathrm { H } _ { 0 } : \sigma _ { 1 } = \sigma _ { 2 }$ , is true, what is the distribution of this test statistic?

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 23 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.995$ to its right.

A sample standard deviation and sample size are given. Use the one-standard-deviation $x ^ { 2 }$ -test to conduct the requiredhypothesis test. - $\mathrm { s } = 979 , \mathrm { n } = 22$ $\mathrm { H } _ { 0 } : \sigma = 650 , \mathrm { H } _ { \mathrm { a } } : \sigma > 650 , \alpha = 0.01$

$\text { Use the chi-square table to find the required } \chi ^ { 2 } \text {-value(s). }$ -For a $\chi ^ { 2 }$ -curve with 25 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.005$ to its right.