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Problem 46 Hard Difficulty

About 28$\%$ of private companies are owned by women (The Cincinnati Enquirer, January $26,2006 ) .$ Answer the following questions based on a sample of 240 private companies.
a. Show the sampling distribution of $\overline{p},$ the sample proportion of companies that are owned by women.
b. What is the probability the sample proportion will be within $\pm .04$ of the population proportion?
c. What is the probability the sample proportion will be within $\pm .02$ of the population proportion?


a. See graph
b. 0.8324
c. 0.5098


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Video Transcript

all right. This question gives us the population proportion of female owned companies, and it wants us to compete some probabilities about this. So party wants the probability that we're within 0.4 of the population mean in a sample of 2 40 So we could rewrite this as the probability that we get a P had value between 0.24 1.32 which, of course, we get just by adding and subtracting 0.4 from are mean. So we can use normal CDF to find this probability. With our lower bound of 0.24 our upper bound of 0.32 are mean of 0.28 and our standard error since we're dealing with a sample of point to a times 0.72 all over 2 40 which is course square root of peak, you over and and this works out to be point a 3 to 5. And of course, we can use normal CDF here because we have a large sample. So our sampling distribution is gonna be approximately normal with a mule at 0.28 and a Sigma of square root 0.28 0.72 over to 40 due to our large counts condition, because if we have a large enough count, we're guaranteed at least some degree of normality. So no, we can do Part B, which is very similar, except this time it wants that the probability that we're within 0.2 in a sample of 2 40 So in a very similar fashion to the last question this time it's asking what is the probability that our proportion in our sample is between 0.26 in 0.30 which we can, of course, find by normal CDF of our lower bound upper bound I mean and our standard area is the same as it was in the last part because we have the same population, proportion and the same sample size, and this one works out to be 0.50 nine AIDS.