Exercises 8.1-7, 8.1-10, 8.4-1, 8.4-3, 8.4-6 .
8.1-7. Vitamin B6 is one of the vitamins in a multiple vita-min pill manufactured by a pharmaceutical company. The pills are produced with a mean of 50 mg of vitamin B6 per pill. The company believes that there is a deteriora-tion of 1 mg/month, so that after three months it expects that ? = 47. A consumer group suspects that ?< 47 after three months.
(a) Define a critical region to test H0: ? = 47 against H1: ?< 47 at an ? = 0.05 significance level based on a random sample of size n = 20.
(b) If the 20 pills yielded a mean of x = 46.94 with a stan-dard deviation of s = 0.15, what is your conclusion?
(c) What is the approximate p value of this test?
8.1-10. In a mechanical testing lab, acrylic glass strips are stretchedtofailure.Let X equal the change in length in mm before breaking. Assume that the distribution of X is N(?, ?2). We shall test the null hypothesis H0: ? = 5.70 against the alternative hypothesis H1: ?> 5.70, using n = 8 observations of X.
(a) Define the test statistic and a critical region that has a significance level of ? = 0.05. Sketch a figure showing this critical region.
(b) A random sample of eight observations of X yielded the following data: 5.71 5.80 6.03 5.87 6.22 5.92 5.57 5.83
Calculate the value of the test statistic and state your conclusion clearly.
(c) Give the approximate value of or bounds for the p-value of this test.
8.4-1. Let Y be b(100, p). To test H0: p = 0.08 against H1: p < 0.08, we reject H0 and accept H1 if and only if Y ? 6. (a) Determine the significance level ? of the test. (b) Find the probability of the Type II error if, in fact, p = 0.04.
8.4-3. Let Y be b(192, p). We reject H0: p = 0.75 and accept H1: p > 0.75 if and only if Y ? 152. Use the normal approximation to determine (a) ? = P(Y ? 152; p = 0.75). (b) ? = P(Y < 152) when p = 0.80.
8.4-6. It was claimed that 75% of all dentists recommend a certain brand of gum for their gum-chewing patients. A consumer group doubted this claim and decided to testH0: p = 0.75 against the alternative hypothesis H1: p < 0.75, where p is the proportion of dentists who recommend that brand of gum. A survey of 390 dentists found that 273 recommended the given brand of gum.
(a) Which hypothesis would you accept if the significance level is ? = 0.05?
(b) Which hypothesis would you accept if the significance level is ? = 0.01?
(c) Find the p-value for this test.