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 the sample is anywhere from 6 to 12, we shall not reject the null hypothesis that p = 0.6; otherwise, we shall conclude that p �/= 0.6.

Evaluate β for the alternatives p = 0.5 and p = 0.7.

Beta is The probability of Type 2 error.
I tried to use the binomial distribution with n=15 and p=.5 and .7 to get P(6<=x<=12|p=.5) and use 12 and 6 for x and substract the results but I can't get the correct answers which are 0.8454 for p=0.5, and 0.8695 for p=0.7
Is there another way of doing it? Are my x's right for the binomial distribution right?

To evaluate β for the alternatives p = 0.5 and p = 0.7, you will need to use the binomial distribution and calculate the probability of observing a sample proportion outside the range from 6 to 12.

The formula for calculating the probability in the binomial distribution is:

P(x=k) = nCk * p^k * (1-p)^(n-k)

Where:
- n is the sample size (15 in this case)
- p is the proportion of college graduates in the population (0.5 or 0.7 in the alternatives)
- k is the number of college graduates in the sample (ranging from 0 to 15)

To find β, you will calculate the probability of x being less than 6 or greater than 12 for each alternative hypothesis.

For p = 0.5:
1. Calculate P(x < 6) by summing up the probabilities from x = 0 to x = 5.
2. Calculate P(x > 12) by summing up the probabilities from x = 13 to x = 15.
3. Add both probabilities calculated in steps 1 and 2 to get the total probability outside the range.
4. β, the probability of Type 2 error, is equal to the total probability outside the range.

For p = 0.7:
1. Repeat steps 1-4 for p = 0.7 using the appropriate probabilities.

Ensure that you use the correct values for n, p, and k in the binomial formula and calculate the probabilities accurately to get the correct answers for β.

To evaluate the probability of Type 2 error (β), we need to calculate the probability of rejecting the null hypothesis (concluding that p ≠ 0.6) when the true proportion (p) is either 0.5 or 0.7.

To calculate this probability, we can use the binomial distribution with parameters n (sample size) and p (true proportion). We then calculate the cumulative probability of observing 6 to 12 college graduates in the sample (6 <= X <= 12), assuming each value of p.

Let's calculate β for both alternatives:

1. Alternative 1: p = 0.5
Using the binomial distribution with n = 15 and p = 0.5, we calculate the cumulative probability of observing 6 to 12 successes:
P(6 <= X <= 12 | p = 0.5) = Σ [ P(X = k | p = 0.5) ] for k = 6 to 12

2. Alternative 2: p = 0.7
Using the binomial distribution with n = 15 and p = 0.7, we calculate the cumulative probability of observing 6 to 12 successes:
P(6 <= X <= 12 | p = 0.7) = Σ [ P(X = k | p = 0.7) ] for k = 6 to 12

Now, let's calculate β for each case.