Question

1. Suppose you suspect a coin is biased against tails, and that you decided to conduct a test of hypothesis by tossing the coin n = 15 times.

a. What are the null and the alternative hypotheses?

b. What is the rejection region in terms of X = number of tails obtained in the 15 tosses that you need to use so that the level ? of the test is as close as 0.05 (remember your ? needs to be below 0.05).

c. If the coin you are tossing is truly biased against tails, and has a probability of tails equal to 0.3, what is the probability that you will let the owner of the coin walk scotch free when using the ? level you set in part b above? That is, what is the probability ? that you will make at Type II error?

d. If the coin you are tossing is truly biased against tails, and has a probability of tails equal to 0.1, what is the probability that you will let the owner of the coin walk scotch free when using the ? level you set in part b above? That is, what is the probability ? that you will make at Type II error?

e. If once you tossed the coin 15 times you obtain 6 tails, can you reject the null hypothesis? Why? Interpret your results in terms of the subject matter.

f. What is the p-value of your result in part e?

g. If once you tossed the coin 15 times you obtain 3 tails, can you reject the null hypothesis? Why? Interpret your results in terms of the subject matter.

h. What is the p-value of your result in part g?

i. Suppose you decide to conduct a test so that you have a lower probability of Type I error than that you used in part a above. What is the rejection region in terms of X = number of tails obtained in the 15 tosses that you need to use so that the level ? of the test is as close as 0.01 (remember alpha needs to be below 0.01).

j. If the coin you are tossing is truly biased against tails, and has a probability of tails equal to 0.3, what is the probability that you will let the owner of the coin walk scotch free when using the ? level you set in part i above? That is, what is the probability ? that you will make at Type II error? Compare this value with the one you obtained in part c above and comment on this comparison. k. If the coin you are tossing is truly biased against tails, and has a probability of tails equal to 0.1, what is the probability that you will let the owner of the coin walk scotch free when using the ? level you set in part b above? That is, what is the probability ? that you will make at Type II error? Compare this value with the one you obtained in part d above and comment on this comparison.

Answer 1

1:

(a)

Let p shows the true proportion of tails. Hypotheses are:

(b)

Test is left tail so critical value of z for which we will reject the null hypothesis is -1.645.

Standard deviation of the proportion is:

The critical value of sample proportion is

So required X is

c)

Standard deviation of the proportion is:

The z-score for and p=0.3 is

The type II error is

d)

Standard deviation of the proportion is:

The z-score for and p=0.1 is

The type II error is