Exam 8: Hypothesis Testing

Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim. -The principal of a middle school claims that test scores of the seventh-graders at his school vary less than the test scores of the seventh-graders at a neighboring school, which have variation described by $\sigma = 14.7$ . Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.

Find the value of the test statistic z using z $z = \frac { \hat { p } - p } { \sqrt { \frac { p q } { n } } }$ -A claim is made that the proportion of children who play sports is less than 0.5, and the sample statistics include n = 1899 subjects with 30% saying that they play a sport.

Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim. -A psychologist claims that more than 24 percent of the population suffers from professional problems due to extreme shyness. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is failure to reject the null hypothesis, state the conclusion in nontechnical terms.

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. -The standard deviation of math test scores at one high school is 16.1. A teacher claims that the standard deviation of the girls' test scores is smaller than 16.1. A random sample of 22 girls results in scores with a standard deviation of 12. Use a significance level of 0.01 to test the teacher's claim.

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion. -Test the claim that the mean age of the prison population in one city is less than 26 years. Sample data are summarized as $\mathrm { n } = 25 , \overline { \mathrm { x } } = 24.4$ years, and $\mathrm { s } = 9.2$ years. Use a significance level of $\alpha = 0.05$ .

Use the given information to find the P-value. Also, use a 0.05 significance level and state the conclusion about the null hypothesis (reject the null hypothesis or fail to reject the null hypothesis). -The test statistic in a left-tailed test i $z = - 1.83$

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. -Heights of men aged 25 to 34 have a standard deviation of 2.9. Use a 0.05 significance level to test the claim that 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. Round the sample standard deviation to five decimal places. 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 traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. -Systolic blood pressure levels for men have a standard deviation of 19.7 mm Hg. A random sample of 31 women resulted in blood pressure levels with a standard deviation of 23.7 mm Hg. Use a 0.05 significance level to test the claim that blood pressure levels for women have the same variation as those for men.

Provide an appropriate response -Test the claim that for the population of history exams, the mean score is 80 . Sample data are summarized as $\mathrm { n } = 16 , \overline { \mathrm { x } } = 84.5$ , and $\mathrm { s } = 11.2$ . Use a significance level of $\alpha = 0.01$ .

Assume that the data has a normal distribution and the number of observations is greater than fifty. Find the critical z value used to test a null hypothesis. - $\alpha = 0.09$ for a right-tailed test.

The principal of a middle school claims that test scores of the seventh-graders at her school vary less than the test scores of seventh-graders at a neighboring school, which have variation described by $\sigma = 14.7$ .

Express the null hypothesis and the alternative hypothesis in symbolic form. Use the correct symbol $( \mu , \mathrm { p } , \sigma )$ for the indicated parameter. -A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO, p, is less than 2 in every one thousand.

Provide an appropriate response. -Complete the following table on hypothesis testing. Test about Distribution Assumptions Mean Mean Proportion Variance

Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense. Last year an appliance manufacturer received many complaints about the high rate of defects among its washing machines. Approximately 9% of the machines were defective in some way. This year the company tightened up its quality control procedures. The latest shipment of 250 washing machines contained 2 defective units.

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. -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. Round the sample standard deviation to three decimal places. 115 128 107 109 116 124 135 127 115 104 118 126 129 133

In a hypothesis test, which of the following will cause a decrease in $\beta$ , the probability of making a type II error? A: Increasing $\alpha$ while keeping the sample size $n$ , fixed B: Increasing the sample size $n$ , while keeping $\alpha$ fixed $\mathrm { C }$ : Decreasing $\alpha$ while keeping the sample size $n$ , fixed D: Decreasing the sample size $n$ , while keeping $\alpha$ fixed

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim -A researcher wants to test the claim that convicted burglars spend an average of $18.7$ months in jail. She takes a random sample of 11 such cases from court files and finds that $\bar { x }$ $= 21.3$ months and $\mathrm { s } = 7.7$ months. Test the claim that $\mu = 18.7$ months at the $0.05$ significance level. Use the traditional method of testing hypotheses.

For large numbers of degrees of freedom, the critical $\chi ^ { 2 }$ values can be approximated as follows: $\chi ^ { 2 } = \frac { 1 } { 2 } ( \mathrm { z } + \sqrt { 2 \mathrm { k } - 1 } ) ^ { 2 }$ where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z }$ is the critical value. To find the lower critical value, the negative z-value is used, to find the upper critical value, the positive z-value is used. Use this approximation to estimate the critical value of $\chi ^ { 2 }$ in a two-tailed hypothesis test with $\mathrm { n } = 146$ and $\alpha = 0.10$ .

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. -List three phrases which are associated with one-tailed claims.

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim. -Use a significance level of $\alpha = 0.05$ to test the claim that $\mu = 32.6$ . The sample data consist of 15 scores for which $\bar { x } = 41.6$ and $s = 8$ . Use the traditional method of testing hypotheses.