Exam 10: Hypothesis Tests Regarding a Parameter

Distinguish between Statistical Significance and Practical Significance -True or False: Results that are practically significant will always be statistically significant.

State Conclusions to Hypothesis Tests -The dean of a major university claims that the mean number of hours students study at her University (per day) is at most $3.8$ hours. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis?

Test Hypotheses about a Population Mean with Known Using P-values -Suppose you are using $\alpha = 0.05$ to test the claim that $\mu > 24$ using a P-value. You are given the sample statistics $n = 50 , \bar { x } = 24.3$ , and $\sigma = 1.2$ . Find the P-value.

Provide an appropriate response. -The mean repair bill of cars is greater than $110. Write the existing state and alternative hypotheses.

Test Hypotheses about a Population Proportion -Compute the standardized test statistic, $\chi ^ { 2 }$ , to test the claim $\sigma ^ { 2 } \leq 9.6$ if $\mathrm { n } = 20 , \mathrm {~s} ^ { 2 } = 18.6$ , and $\alpha = 0.01$ .

Test Hypotheses about a Population Proportion -In a hypothesis test as the population mean gets closer to the hypothesized mean, $\beta$

Test Hypotheses about a Population Mean with Known Using P-values -If the level of significance is $0.05$ , and the $\mathrm { P }$ -value is $0.043$ , the decision would be to

Explain Type I and Type II Errors -A local tennis pro-shop strings tennis rackets at the tension (pounds per square inch) requested by the customer. Recently a customer made a claim that the pro-shop consistently strings rackets at lower tensions, on average, than requested. To support this claim, the customer asked the pro shop to string 16 new rackets at 50 psi. Upon receiving the rackets, the customer measured the tension of each and calculated the following summary statistics: $\bar { x } = 49 \mathrm { psi } , \mathrm { s } = 4.0 \mathrm { psi }$ . In order to conduct the test, the customer selected a significance level of $\alpha = .01$ . Interpret this value.

Test Hypotheses about a Population Proportion -An event is considered unusual if the probability of observing the event is A) less than $0.05$ B) less than $0.025$ C) less than $0.10$ D) greater than $0.95$

Test Hypotheses about a Population Mean with Known Using the Classical Approach -A shipping firm suspects that the mean lifetime of the tires used by its trucks is less than 40,000 miles. To check the claim, the firm randomly selects and tests 54 of these tires and gets a mean lifetime of 39,370 miles with a standard deviation of 1200 miles. At $\alpha = 0.05$ , test the shipping firm's claim.

Test Hypotheses about a Population Mean with Unknown -Find the standardized test statistic t for a sample with $\mathrm { n } = 12 , \overline { \mathrm { x } } = 18.5 , \mathrm {~s} = 2.1$ , and $\alpha = 0.01$ if $\mathrm { H } _ { 1 } : \mu \neq 19$ . Round your answer to three decimal places.

Explain the Logic of Hypothesis Testing -Find the critical value for a two-tailed test with $\alpha = 0.10$ .

Test Hypotheses about a Population Proportion -A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in $25 \%$ of the women who actually have the disease. A new method has been developed that researchers hope will be able to detect cancer more accurately. A random sample of 70 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 8 . Is the sample size sufficiently large in order to conduct this test of hypothesis? Explain.

Test Hypotheses about a Population Mean with Unknown - $\text { Use a t-test to test the claim } \mu \geq 13 \text { at } \alpha = 0.05 \text {, given the sample statistics } \mathrm { n } = 10 , \overline { \mathrm { x } } = 12.1 \text {, and } \mathrm { s } = 1.3 \text {. }$

Explain Type I and Type II Errors -The mean cost of textbooks for one class is greater than $160. Identify the type I and type II errors for the hypothesis test of this claim.

Test Hypotheses about a Population Mean with Unknown -If we have a sample of 12 drawn from a normal population, then we would use as our test statistic

Explain Type I and Type II Errors -If we do not reject the null hypothesis when the null hypothesis is in error, then we have made a

Test Hypotheses about a Population Proportion -Test the claim that $\sigma < 21.33$ if $\mathrm { n } = 28 , \mathrm {~s} = 16.83$ and $\alpha = 0.10$ . Assume that the population is normally distributed.

Test Hypotheses about a Population Proportion -An airline claims that the no-show rate for passengers is less than $5 \%$ . In a sample of 420 randomly selected reservations, 19 were no-shows. At $\alpha = 0.01$ , test the airline's claim.

Test Hypotheses about a Population Mean with Known Using P-values -Given $\mathrm { H } _ { 0 } : \mu \geq 18 , \mathrm { H } _ { 1 } : \mu < 18$ , and $\mathrm { P } = 0.070$ . Do you reject or fail to reject $\mathrm { H } _ { 0 }$ at the $0.05$ level of significance?