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. She performs a right-tailed chi-square test using a random sample of $10$ adults. She 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.