Exam 10: Hypothesis Tests Regarding a Parameter

Test Hypotheses about a Population Mean with Known Using P-values -What is a P-value?

Test Hypotheses about a Population Proportion -A recent study claimed that at least $15 \%$ of junior high students are overweight. In a sample of 160 students, 18 were found to be overweight. At $\alpha = 0.05$ , test the claim.

Test Hypotheses about a Population Proportion -A statistics professor at an all-men's college determined that the standard deviation of men's heights is $2.5$ inches. The professor then randomly selected 41 female students from a nearby all-female college and found the standard deviation to be $2.9$ inches. Test the professor's claim that the standard deviation of female heights is greater than $2.5$ inches. Use $\alpha = 0.01$ .

Provide an appropriate response. -The owner of an outdoor store recommends against buying the new model of one brand of GPS receivers because they vary more than the old model, which had a standard deviation of 50 meters. Write the null and alternative hypotheses.

Test Hypotheses about a Population Proportion -A survey claims that 9 out of 10 doctors (i.e., $90 \%$ ) recommend brand $Z$ for their patients who have children. To test this claim against the alternative that the actual proportion of doctors who recommend brand $Z$ is less than $90 \%$ , a random sample of doctors was taken. Suppose the test statistic is $\mathrm { z } = - 1.95$ . Can we conclude that $\mathrm { H } _ { 0 }$ should be rejected at the a) $\alpha = 0.10$ , b) $\alpha = 0.05$ , and c) $\alpha = 0.01$ level?

Test Hypotheses about a Population Proportion -Fifty-five percent of registered voters in a congressional district are registered Democrats. The Republican candidate takes a poll to assess his chances in a two-candidate race. He polls 1200 potential voters and finds that 621 plan to vote for the Democratic candidate. Does the Republican candidate have a chance to win? Use $\alpha$ $= 0.05$ .

Test Hypotheses about a Population Mean with Unknown -A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful if the average time spent on the deliveries does not exceed 30 minutes. The owner has randomly selected 16 customers and has delivered pizzas to their homes in order to test if the mean delivery time actually exceeds 30 minutes. Suppose the P-value for the test was found to be $0.0287$ . State the correct conclusion.

Test Hypotheses about a Population Mean with Unknown -Find the standardized test statistic t for a sample with n = 10, x = 8.8, s = 1.3, and = 0.05 if H0: 9.7. Round your answer to three decimal places.

Test Hypotheses about a Population Mean with Known Using P-values -Given $\mathrm { H } _ { 0 } : \mu = 25 , \mathrm { H } _ { 1 } : \mu \neq 25$ , and $\mathrm { P } = 0.035$ . Do you reject or fail to reject $\mathrm { H } _ { 0 }$ at the $0.01$ level of significance?

Test Hypotheses about a Population Proportion -Compute the standardized test statistic, $\chi ^ { 2 }$ to test the claim $\sigma ^ { 2 } \neq 54.4$ if $\mathrm { n } = 10$ , $\mathrm { s } ^ { 2 } = 60$ , and $\alpha = 0.01$ .

Test Hypotheses about a Population Proportion -Compute the standardized test statistic, $\chi ^ { 2 }$ , to test the claim $\sigma ^ { 2 } \geq 14.4$ if $\mathrm { n } = 15$ , $\mathrm { s } ^ { 2 } = 12$ , and $\alpha = 0.05$ .

Test Hypotheses about a Population Proportion -It has been estimated that the $1991 \mathrm { G }$ -car obtains a mean of 35 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects $361991 \mathrm { G }$ -cars and records the mileage obtained for each car over a driving course similar to that used to obtain the estimate. The following data resulted: $\bar { x } = 36.5$ miles per gallon, $\mathrm { s } = 6$ miles per gallon. Calculate the value of $\beta$ if the true value of the mean is really 37 miles per gallon. Use $\alpha = 0.025$ .

Explain Type I and Type II Errors -If we reject the null hypothesis when the null hypothesis is true, then we have made a

Test Hypotheses about a Population Proportion -Test the claim that $\sigma ^ { 2 } \geq 12.6$ if $\mathrm { n } = 15 , \mathrm {~s} ^ { 2 } = 10.5$ , and $\alpha = 0.05$ . Assume that the population is normally distributed.

Test Hypotheses about a Population Mean with Unknown - $\text { Use a t-test to test the claim } \mu = 24 \text { at } \alpha = 0.01 \text {, given the sample statistics } \mathrm { n } = 12 , \overline { \mathrm { x } } = 25.2 \text {, and } \mathrm { s } = 2.2 \text {. }$

Explain Type I and Type II Errors -The level of significance, $\alpha$ , is the probability of making a

Test Hypotheses about a Population Mean with Known Using the Classical Approach -In a two-tailed test of a hypothesis using the classical method, the critical value is denoted by

Test Hypotheses about a Population Proportion -Determine the standardized test statistic, $z$ , to test the claim about the population proportion $p > 0.015$ given $\mathrm { n } = 150$ and $\hat { \mathrm { p } } = 0.027$ . Use $\alpha = 0.01$ .

Explain Type I and Type II Errors -True or False: If I specify $\beta$ to be equal to $0.34$ , then the value of $\alpha$ must be $0.66$ .

Explain Type I and Type II Errors -The mean monthly cell phone bill for one household was less than $109. Identify the type I and type II errors for the hypothesis test of this claim.