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.
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 12 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
To evaluate β for the alternatives p = 0.5 and p = 0.7, we need to calculate the probability of rejecting the null hypothesis when it is false. In this case, rejecting the null hypothesis means observing a number of college graduates in the sample of less than 6 or greater than 12.
To find this probability, we can use the binomial distribution formula. Let's break down the steps to calculate it:
1. Calculate the probability of observing 6 or more college graduates in the sample when p = 0.5:
P(6<=x<=12 | p = 0.5) = P(x <= 12 | p = 0.5) - P(x <= 5 | p = 0.5)
To calculate P(x <= 12 | p = 0.5), we can use a binomial cumulative distribution function (cdf) or calculate each individual probability and sum them up. Similarly, we can calculate P(x <= 5 | p = 0.5). Let's calculate using individual probabilities:
P(x <= 12 | p = 0.5) = P(x=0) + P(x=1) + P(x=2) + ... + P(x=12)
P(x <= 5 | p = 0.5) = P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4) + P(x=5)
We can use the binomial distribution formula, which involves the number of trials (n), the probability of success (p), and the specific value of x. In this case, n=15 and p=0.5.
P(x=k) = (n choose k) * p^k * (1 - p)^(n - k)
Using this formula, we can calculate each individual probability and sum them up. Let's calculate using a programming language or a tool with binomial distribution capabilities such as a statistical software package.
Calculating P(x <= 12 | p = 0.5) and P(x <= 5 | p = 0.5) in this manner should give you the correct probability value for p = 0.5, which is 0.8454.
2. Repeat the same steps for p = 0.7:
P(6<=x<=12 | p = 0.7) = P(x <= 12 | p = 0.7) - P(x <= 5 | p = 0.7)
Use the binomial distribution formula, plugging in n=15 and p=0.7 to calculate each individual probability and sum them up. Again, programming languages or statistical software packages can help with these calculations.
Calculating P(x <= 12 | p = 0.7) and P(x <= 5 | p = 0.7) in this manner should give you the correct probability value for p = 0.7, which is 0.8695.
Remember to subtract P(x <= 5 | p) from P(x <= 12 | p) to get the probability of rejecting the null hypothesis.