Question

Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing H₀: p = .5versus H_a: p ≠ .5 based on a random sample of 25 individuals. Let X denote the number in the sample who favor the first company and x represent the observed value of X.

a. Which of the following rejection regions is most appropriate and why?

b. In the context of this problem situation, describe what the type I and type II errors are.

c. What is the probability distribution of the test statistic X when H₀ is true? Use it to compute the probability of a type I error.

d. Compute the probability of a type II error for the selected region when p = .3 again when p = .4 and also for both p = .6 and p = .7

e. Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?

Answer 1

Let be the number in the sample that favours the first company over the second.

Let be the proportion of all potential subscribers who favour the first company over the second.

Let the sample size be.

The null and alternative hypotheses are,

(a)

Since the alternate hypothesis is . So, the rejection region should be two tailed thus is most appropriate.

(b)

Type I error: Reject the null hypothesis when it is true.

Here, it can be judge that a majority favour one of the two companies when that is not the case.

Type II error: Fail to reject the null hypothesis when it false.

Here, it can be judge that potential subscribers are evenly split between the two companies when they are not.

(c)

When is true the test statistics X has Binomial probability distribution.

That is .

Therefore, the probability of type I error is

(d)

Compute type II error when against when

Here, reject the null hypothesis when or .

When the type II error is,

When the type II error is,

When the type II error is,

When the type II error is,

(e)

Using the results obtained in part (d), the value lies in rejection region. So, the null hypothesis is rejected. In other words, reject the null hypothesis in favour of alternative hypothesis .