9. Hypothesis Testing for One Sample

Finding Critical Values and Rejection Regions In Exercises 23–28, find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer.

Right-tailed test, α = 0.08

Mark ‘TRUE’ or ‘FALSE’ for each of the following.

$\left(A\right)$ The test statistic & the critical value are the same thing.

Mark ‘TRUE’ or ‘FALSE’ for each of the following.

$\left(B\right)$ The critical value is the boundary of the rejection region.

Mark ‘TRUE’ or ‘FALSE’ for each of the following.

$\left(C\right)$ You should always reject $H_0$ if the test statistic is greater than the critical value.

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.

$\left(A\right)$ $H_0:\mu =7.5$; $H_a:\mu >7.5$

$\alpha =0.01;z=2.17$

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.

$\left(B\right)$ $H_0:p=0.64$;

$\alpha =0.10;z=-1.53$

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.

$\left(C\right)$ $H_0:\mu =98.6$; $H_a:\mu \ne 98.6$

$\alpha =0.05$

$t=2.41;df=17$

Use α to find the critical value(s), then determine if the given test statistic is in the rejection region.

$\left(D\right)$ $H_0:{\sigma }^2=0.3;H_a:{\sigma }^2>0.3$

$\alpha =0.05$

${\chi }^2=61.7;df=51$

A soup company claims that the average sodium content of their most popular soup is 500 mg per can. A nutritionist collects a sample of 36 cans with mean sodium content 507 mg. Assume a known pop. standard deviation of 15 mg & test the nutritionist’s suspicion that the mean sodium content is more than 500 mg using the critical value method with $\alpha =0.01$.

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 14

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 15

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 16

Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

a. Use the critical value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

Test Statistic and Critical Value The statistics for the sample data in Exercise 1 are n = 15, x_bar = 6.133333, and s = 8.862978, where the units are millions of dollars. Find the test statistic and critical value(s) for a test of the claim that the salaries are from a population with a mean greater than 5 million dollars. Assume that a 0.05 significance level is used.

To test H0: μ = 100 versus H1: μ ≠ 100, a simple random sample of size n = 23 is obtained from a population that is known to be normally distributed.

a. If x̄ = 104.8 and s = 9.2, compute the test statistic.

b. If the researcher decides to test this hypothesis at the α = 0.01 level of significance, determine the critical values.