A researcher wants to test the claim that the population proportion is greater than $0.45$. The sample proportion is $0.50$, based on a sample of size $50$. Using a significance level of $\alpha =0.05$, should the researcher reject the null hypothesis or fail to reject it? Find the critical $z$-value and determine the conclusion of the hypothesis test.

A sample of $12$ college students has an average monthly rent of $\mathrm{\backslash (}850$ with a sample standard deviation of $\mathrm{\backslash )}120$. Test the claim that the average rent is greater than $\$800$ at the $0.05$ significance level. What are the test statistic and the critical value for this right-tailed test?

A nutritionist is testing whether the variability in daily calcium intake among adults has increased compared to the known standard deviation. The population is assumed to be normally distributed. The nutritionist performs a right-tailed chi-square test using a random sample of $10$ adults. The nutritionist chooses a significance level of $0.10$. What is the critical value and the rejection region for this right-tailed chi-square test?

A one-tailed $t$-test is performed at the $0.025$ significance level with a sample size of $24$. What is the critical $t$-value?

In a right-tailed hypothesis test with $\alpha =0.05$, the calculated $p$-value is $0.02$. Where does the standardized test statistic lie relative to the critical value? Choose the best answer.