statistics question?

Question

management thinks that 45 % of current stockholders will want to make a purchase. A random sample of 130 stockholders is selected, 63 of whom express a desire to buy.
(a) what is the standard error of the sample proportion?
(b)what is the mean of the sampling distribution of sample proportion?
(c) what is the probability of obtaining the results described in the problem if p = .45?

seem to have gotten rusty over the holiday, don't remember how to do this question. help is appreciated.

Answer 1

The standard error is a measure of how far the true mean of a population the mean of a sample of size N can be expected to be:
http://en.wikipedia.org/wiki/Standard_error_(statistics)

To compute the standard error of a sample, you need to know (or have a good estimate for) the variance of the underlying population.

In this case, it seems that you model of the population is that each shareholder makes an independent binary decision of whether to buy or not with probability p for buying. This corresponds to a binomial distribution:
http://en.wikipedia.org/wiki/Binomial_distribution

For a single Bernoulli trial, the variance is p(1-p). The formula for standard error is:

E = S/sqrt(N)

where:
E is the standard error
S is the standard deviation of the underlying population
N is the sample size

You have N and the formula for variance in terms of p so you next need to determine p. The best estimate of the "true" p for the population is the data from the sample, in this case 63/130.

This addresses part a.

There are two approaches to part c.

If we were to take a large number of random samples of 130 stockholders and determine the number wanting to buy, we would get a distribution that, to all intents and purposes, would be normal. The standard error would be the standard deviation of the resulting ratios.

So we can compute the standard error using p = 0.45 and determine how many standard deviations away (63/130) is from 0.45. This give you the approximate probability for part c.

If you want the exact probability, then you have to go to the binomial distribution's probability mass function (see reference above) with p = 0.45, n = 130, and k = 63. But while it is theoretically possible to compute this exactly, no one in his right mind would. One normally uses the approximation associated with the normal distribution.