A random sample of $52$ observations has a sample mean of $2,430$. Assume the population is normally distributed with a known standard deviation of $\sigma =275$. At the $0.05$ level of significance, test the claim that the population mean $\mu$ is not equal to $2,500$.

A snack company produces potato chips. The company claims that the mean weight of a bag is $50$ grams. To monitor quality, a random sample of $16$ bags is selected, and the mean weight is found to be $51$ grams with a standard deviation of $0.4$ grams. Assume the weights are approximately normally distributed. Using a significance level of $\alpha =0.05$ (two-tailed), are the bags meeting the claimed mean weight? (Critical $t$-values for $df=15$ is approximately $\pm 2.131$)

A relationship study claims that $18\%$ of couples have initiated marriage counseling. A random sample of $400$ couples found that $80$ had initiated counseling. Construct a $95\%$ confidence interval for the population proportion and state whether the study's claim of $18\%$ is consistent with the sample data.

A juice company claims the average vitamin C content in its product is less than $60$ mg per serving. After performing a hypothesis test, you reject the null hypothesis. What is the correct interpretation?

A company claims that the mean weight of its protein bars is $50$ grams. A quality control analyst samples 8 bars and obtains a sample mean of $51.2$ grams. After simulating $2000$ randomizations under the null hypothesis, $120$ simulated means are at least as large as $51.2$ grams. What is the correct interpretation of the $p$-value and the decision at $\alpha =0.05$?