1

Let X equal the thickness of spearmint gum manufactured for vending machines. Assume that the distribution of X is N(µ,σ2 ). The target thickness is 7.5 hundredths of an inch. We shall test the null hypothesis, using ten observations.

(a) Define the test statistic and rejection region for an α = 0.05 significance level test.  Sketch a figure

illustrating this critical region.

(b) Calculate the value of the test statistic and state your decision clearly, using the following n = 10

observations, randomly selected from the production line.

7.65    7.60    7.70    7.55    7.65

7.55    7.40    7.40    7.50    7.50

(c) Is our hypothesized mean of µ0  = 7.50 contained in a 95% confidence interval for µ?

2

An experimenter has prepared a drug dosage level that she claims will induce sleep for 80% of people suffering from insomnia. After examining the dosage, we feel that her claims regarding the effectiveness of the dosage are inflated. In an attempt to disprove her claim, we administer her prescribed dosage to 20 insomniacs and we observe Y , the number for whom the drug dose induces sleep. We wish to test the hypothesis H0  : p = 0.8 vs. H1  : p < 0.8. Assume that the rejection region {y ≤ 12} is used.

(a) In terms of this problem, what is a type I error?

(b) Find α .

(c) In terms of this problem, what is a type II error?

(d) Find β .

(e) Find β when p = 0.6.

(f) Find β when p = 0.4.

3

A bowl contains two red balls, two white balls, and a fifth ball that is either red or white. Let p denote the probability of drawing a red ball from the bowl.  We shall test the simple null hypothesis H0  : p = 3/5 vs. H1  : p = 2/5. Say we will draw four balls at random, one at a time and with replacement. Let X equal the number of red balls drawn.

(a) Define a rejection region for this test in terms of X .

(b) For the rejection region you’ve defined, find the values of α and β .