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=.5, and 0.8695 for p=.7
Is there another way of doing it? Are me x's right for the binomial distributio right?
To evaluate β for the alternatives p = 0.5 and p = 0.7, you need to calculate the probability of committing a Type 2 error under each alternative hypothesis.
1. For p = 0.5:
Using the binomial distribution with n = 15 and p = 0.5, you need to calculate P(6 ≤ x ≤ 12 | p = 0.5). This represents the probability of observing 6 to 12 college graduates in the sample, assuming p = 0.5.
To calculate this, you can find the individual probabilities for x = 6, 7, 8, 9, 10, 11, and 12, and then sum them up:
P(6 ≤ x ≤ 12 | p = 0.5) = P(x = 6) + P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10) + P(x = 11) + P(x = 12)
You can use a binomial probability calculator or software to directly find these values. Summing them up will give you the probability of observing 6 to 12 college graduates in the sample, assuming p = 0.5.
2. For p = 0.7:
Similarly, using the binomial distribution with n = 15 and p = 0.7, you need to calculate P(6 ≤ x ≤ 12 | p = 0.7) to determine the probability of observing 6 to 12 college graduates in the sample, assuming p = 0.7. Again, find the individual probabilities for x = 6, 7, 8, 9, 10, 11, and 12, and sum them up:
P(6 ≤ x ≤ 12 | p = 0.7) = P(x = 6) + P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10) + P(x = 11) + P(x = 12)
Once you have the probabilities for both alternatives, you can subtract them from 1 to find β (the probability of a Type 2 error) for each alternative hypothesis.
Note: It's important to ensure that you are using the correct formulas and calculations for the binomial probabilities. Double-check the formulas and calculations you are using, or consider using statistical software or online calculators for accurate results.
To evaluate β, the probability of a Type 2 error, we need to calculate the probability of rejecting the null hypothesis (concluding that p ≠ 0.6) when it is actually false. In this case, we are considering two alternatives, p = 0.5 and p = 0.7.
To calculate β for p = 0.5, we need to find the probability of observing a sample proportion (x) between 6 and 12 (inclusive) given that p = 0.5. We can use the binomial distribution:
P(6 ≤ X ≤ 12 | p = 0.5) = P(X ≤ 12 | p = 0.5) - P(X ≤ 5 | p = 0.5)
To calculate this, you need to sum the probabilities of getting 0 through 12 college graduates out of 15, using the binomial distribution formula with p = 0.5.
Similarly, to calculate β for p = 0.7, you need to find the probability of observing a sample proportion between 6 and 12 (inclusive) given that p = 0.7:
P(6 ≤ X ≤ 12 | p = 0.7) = P(X ≤ 12 | p = 0.7) - P(X ≤ 5 | p = 0.7)
Again, use the binomial distribution formula with p = 0.7 to calculate this.
Once you have calculated both probabilities, subtract them from 1 to find β:
β = 1 - P(6 ≤ X ≤ 12 | p = 0.5) for p = 0.5
β = 1 - P(6 ≤ X ≤ 12 | p = 0.7) for p = 0.7
Make sure you are using the correct binomial distribution formula and calculating the probabilities accurately to obtain the correct answers.