A drug manufacturer produces bb pills with a measure, x, called a hardness factor (HF). It is known that x has a normal distribution, and its standard deviation is unknown. It is desired that x has a mean of at least 11.45. A random sample of 12 bb pills had the following hardness factors:
11.63, 11.37, 11.38, 11.48, 11.61, 11.59, 11.72, 11.57, 11.49, 11.46, 11.49, 11.62
So, we assume that our sample comes from a normal population with unknown standard deviation of 𝜎. We would like to test whether the mean HF is higher than 11.45. The null hypothesis is this H0: 𝜇=11.45. We will test this against the alternative Ha. We want to test at 2% level. Let x-bar = the sample mean and s = the sample standard deviation.
(a) What should the alternative hypothesis, Ha, be?
(b) What is the formula for your test statistic?
(c) What value does your test statistic, T, take on with the sample data?
(d) What type of probability distribution does your test statistic, T, have?
(e) How many degrees of freedom does T have?
(f) Calculate the critical value, tstar, for your test (positive value)
(g) For what values of your test statistic, T, is the bull hypothesis rejected?
(h) Calculate the p-value for this test
(I) Is the null hypothesis rejected?
(j) If we ran 900, 2% level tests then about how many times would we make a Type I error?
(k) Create a 98% confidence interval for the mean hardness factor of bb pills based on this sample
(l) Write out R script or any other comments for the above