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?
Suppose that the antenna lengths of woodlice are approximately normally distributed with a mean of 0.2 inches and a standard deviation of 0.05 inches. What proportion of woodlice have antenna lengths that are more than 0.23 inches? Round your answer to at least four decimal places.
To evaluate β for the alternatives p = 0.5 and p = 0.7, you are correct in using the binomial distribution. However, when calculating the probabilities, it's important to consider the two-tailed nature of the test.
To find β, which represents the probability of a Type 2 error, we need to calculate the probability of rejecting the null hypothesis when it is actually false. In this case, the null hypothesis is p = 0.6, and the alternative hypotheses are p = 0.5 and p = 0.7.
To calculate β for p = 0.5:
1. Calculate the cumulative probability of the lower bound (x = 6) using the binomial distribution with n = 15 and p = 0.5. Let's call this probability P1.
2. Calculate the cumulative probability of the upper bound (x = 12) using the same parameters. Let's call this probability P2.
3. Calculate β by subtracting P1 from 1 and then subtracting P2 from that result.
To calculate β for p = 0.7:
1. Repeat steps 1 and 2 as described for p = 0.5, but this time using the value of p = 0.7.
2. Calculate β by subtracting P1 from 1 and then subtracting P2 from that result.
Using these steps, let's calculate β for each alternative:
For p = 0.5:
1. P1 = P(X ≤ 6) = sum of P(X = 6), P(X = 5), ..., P(X = 0) using the binomial distribution with n = 15 and p = 0.5.
2. P2 = P(X ≥ 12) = sum of P(X = 12), P(X = 13), ..., P(X = 15) using the same parameters.
3. β = 1 - P1 - P2
For p = 0.7:
1. P1 = P(X ≤ 6) = sum of P(X = 6), P(X = 5), ..., P(X = 0) using the binomial distribution with n = 15 and p = 0.7.
2. P2 = P(X ≥ 12) = sum of P(X = 12), P(X = 13), ..., P(X = 15) using the same parameters.
3. β = 1 - P1 - P2
By following these steps, you should be able to correctly calculate β for the given values of p.