A company claims that at least $60\%$ of its customers are satisfied with its product. To test this claim, a sample of $80$ customers is surveyed, and it is found that $55\%$ of the sample reports being satisfied. Using a significance level of $\alpha =0.05$, test the claim by calculating the critical $z$-value, and determine whether to reject or fail to reject the null hypothesis.

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?

Determine the critical value(s) and rejection region(s) for a two-tailed $z$-test with $\alpha =0.08$.

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?