In a sample of 500 students 50 percent attend college within 50km of their homes The probability that the population proportion will be between 0.45 and 0.55 is?

Answer 1

According to the theory behind a sampling distribution of a proportion, when you take a sample proportion with mean p from a sample of n people, the actual population proportion will follow a normal distribution of mean p with a standard deviation of √(p*(1-p)/n). Using the information given, our sample had a mean, p, of .5 and a sample size, n, of 500. Therefore, the mean of the population is .5 and the standard deviation is √(.5*(1-.5)/500)=.022361.

Next, in order to find our probability, we need to calculate the z-scores of our 2 bounds using the formula z=(x-mean)/standard deviation. For .45 this gives (.45-.5)/.022361=-2.236 and for .55 we get (.55-.5)/.022361=2.236. In order to convert this into a probability, we will need to look these values up in a z-table and find the area between them. Doing that we find that the area must be .974653. This tells us that the probability that the population proportion is between 0.45 and 0.55 is 97.4653%.