Question

The proportion of adults living in a small town who are college graduates is estimated to be p = 0.6. To test this hypothesis, a random sample of 15 adults is selected. If the number of college graduates in our sample is anywhere from 6 to 12, we shall not reject the null hypothesis that p = 0.6; otherwise, we shall conclude that pstudent submitted image, transcription available below0.6. a) Evaluate α assuming that p = 0.6. Use the binomial distribution b) Evaluate β for the alternatives p = 0.5 and p = 0.7 c) Is this a good test procedure? d) Repeat excersize when 200 adults are selected and the fail to reject region is defined to bestudent submitted image, transcription available below, where x is the number of college graduates in our sample. Use the normal approximation.

Answer 1

a) α (Type I error) assuming that p = 0.6 is approximately 0.0609 . This is the probability of incorrectly rejecting the null hypothesis when it is true.

b) β (Type II error) for the alternatives:

- For p = 0.5 , β is approximately 0.8454 . This is the probability of incorrectly accepting the null hypothesis when the true proportion is 0.5.

- For p = 0.7 , β is approximately 0.8695 . This is the probability of incorrectly accepting the null hypothesis when the true proportion is 0.7.

c) Evaluation of the Test Procedure:

- The value of α is quite low, indicating a small chance of a Type I error.

- However, the values of β, especially for p = 0.5 and p = 0.7 are quite high.

- A good test should balance α and β, but in this case, β is quite high, which might be a concern. It suggests the test is not very sensitive to detecting deviations from p = 0.6

d) For the Scenario with 200 Adults:

n = 200 , the binomial distribution can be approximated using a normal distribution:

Normal Approximation:

`\[ \mu = np \]`

`\[ \sigma = √(np(1-p)) \]`

To answer these questions, we'll use statistical concepts and calculations related to hypothesis testing, binomial distribution, and normal approximation.

a) Evaluate α assuming that p = 0.6

α (Type I error) is the probability of rejecting the null hypothesis when it is true. Here, the null hypothesis
`\( H_0 \) is \( p = 0.6 \).` The rejection region is defined as having fewer than 6 or more than 12 college graduates in the sample of 15. We calculate α using the binomial distribution.

Binomial Distribution Formula:

`\[ P(X = k) = \binom{n}{k} p^k (1 - p)^(n - k) \]`

Where n = 15 , p = 0.6 , and X is the number of college graduates in the sample.

To find α, sum the probabilities of getting fewer than 6 or more than 12 graduates:

`\[ \alpha = P(X < 6) + P(X > 12) \]`

`\[ \alpha = \sum_(k=0)^(5) P(X = k) + \sum_(k=13)^(15) P(X = k) \]`

Let's calculate this.

b) Evaluate β for the alternatives p = 0.5 and p = 0.7

β (Type II error) is the probability of not rejecting the null hypothesis when it is false. We calculate β for p = 0.5 and p = 0.7 . The non-rejection region is 6 to 12 graduates.

For p = 0.5 and p = 0.7 :

`\[ \beta = P(6 \leq X \leq 12) \]`

`\[ \beta = \sum_(k=6)^(12) P(X = k) \]`

We'll calculate this for both p = 0.5 and p = 0.7 .

c) Is this a good test procedure?

We'll evaluate this based on the values of α and β.

d) Repeat exercise with 200 adults and normal approximation

When n = 200 , the binomial distribution can be approximated using a normal distribution:

Normal Approximation:

`\[ \mu = np \]`

`\[ \sigma = √(np(1-p)) \]`