Air- STATISTICS SECTtoN II Part B Question 6 Spend about 25 minutes on this part of the exam. Percent of Section I] grade "\"25 Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. E. A company producing croissants containing wheat starch labels the product as glutenfree and therefore safe for people with gluten allergies. According to the [1.5. Food and Drug Administration {FDA}, food containing wheat starch can only be labeled glutenfree if the product has been processed to remove gluten and tests below 20 parts per million [ppm] of gluten, on average. A quality control manager is investigating whether the gluten content in the croissants exceeds the FDA standard. The manager randoitdy selected Eli] of these croissants, measured the gluten content of each croissant, and will use these data to test the hypotheses Hu:u=2 Hazuhl where n represents the population mean gluten content [in ppm) of' the croissants produced by this company. {a} State the conditions for a onesample ttest and explain why the conditions for inference are considered to be met. (b) The t-test resulted in a p-value of 0.34. Based on the p-value, what conclusion should the quality control manager make at the a = 0.10 significance level?{c} Based on your conclusion in part [b], which type of error, Type I or Type II is possible? Describe the consequence of this error in the context of this study. After the preliminary study, one of the company's executives raised a concern because a histogram of the sample measurements appeared strongly tightskewed, but the assumptions of the .ttest are perfectly satisfied only 1when the sample data come from a normally distributed population. The executive decided to conduct a simulation study to investigate the impact the population distribution has on the power of the test when the true mean is in reality only 2 ppm more than the hypothesized value of ED ppm. {d} The executive rst assumed the 40 measurements were taken fmm a nottnal distribution with a mean of 22 (only.r 2 ppm more than the hypothesized value of 2D ppm}. Cinethousand trials of a simulation were conducted, and a pvalue was calculated for each trial using the same method as was used to obtain the pvalue in part [b]. From the simulated distribution of pvalues, the power of the test when sampling from a normal population with a mean of 22 if the test is performed at the D. H] level of significance was estimated to be DEBS. Explain what the estimated power of [1383 represents in this context. {e} The executive next assumed the 41] measurements were taken from a tightskewed population with a mean of 22. l[Zinethousand trials of a simulation 1were again conducted, with pvalues calculated as before. A histogram of the pvalues is below. Power Bimuhol: Right-Shared anihnn P'roqumny of p-valoa E s E Bimuiuedp-value Use the information in the histogram to estimate the power of the test when sampling from a tightskewed population with a mean of 22 if the test is performed at the D. It] level of signicance. Indicate how you obtained your estimate. {f} Based on the simulation results, when the true mean gluten content of the croissants exceeds the FDA standard for glutenflee labeling by 2 ppm, What impact does the shape of the population distribution have on the power of the test