(a) A fitness company wants to determine whether the Paleo diet is superior to the standard diet. A study reports a 95% confidence interval of (0.53 kg, 0.75 kg) for the difference in mean lean body mass between adults on the Paleo diet and adults on their usual diet. The researchers used the Paleo diet minus standard diet in their calculations. Write a sentence interpreting the confidence interval in the context of the problem. (b) A large university performs a chi-squared goodness of fit test to determine whether the proportion of students with various socio economic status in STEM majors matches the overall proportions at the university. The test has a p-value of 0.0546; at a significance level of a= 0.1 , what do they conclude? State the conclusion using both statistical language and in the context of the problem. (c) A bakery claims that at least 80% of their "Birthday Blast" cupcakes have rainbow sprinkles. A customer, who feels like they're always getting shortchanged on sprinkles, decides to test this claim. The test has a P-value of 0.0138, on the basis of which they don't fail to reject the null hypothesis. State the type of error (Type I or Type II) that the researchers aren't at risk of making; explain in the context of the problem what that error wouldn't mean. (d) Researchers are interested in the proportion of college students in the United States that haven't taken at least one online course. They calculated a 90% confidence interval of (0.49, 0.55). Interpret the confidence interval in the context of the problem.