Statistics HELP

Question

It has been reported that 20.4% of incoming freshmen indicate that they will major in business or a related field. A random sample of 400 incoming college freshmen was asked their preference, and 104 replied that they were considering business as a major. Estimate the true proportion of freshman business majors with 94% confidence. Does your interval contain 20.4?

Answer 1

This question is asking for a 1 sample confidence interval to estimate the population proportion. To do this we will use the formula p-hat +/- z* sqrt ((p-hat)(1-p-hat)/n. Here are the steps:

1) Determine p-hat by using the formula x/n, where x is the number of "successes" and n is the sample size. Here we have 104 successes out of a sample size of 400.

p-hat = x/n = 104/400 = 0.26.

2) Determine the z*. To determine z* you can look in a z table or use a graphing calculator. To use a graphing calculator we have to first determine alpha/2 (because confidence intervals are 2-tailed):

alpha = 1- Confidence Level = 1-0.94 = 0.06

alpha/2 = 0.06/2 = 0.03

In the calculator, use invNorm (0.03, 0, 1) = -1.88 (we'll use the absolute value for the confidence interval).

3) Plug everything in to your confidence interval formula:

p-hat +/- z* sqrt ((p-hat)(1-p-hat)/n

Lower Bound = 0.26 - 1.88 * sqrt((0.26)(1-0.26)/400) = 0.22

Upper Bound = 0.26 + 1.88 * sqrt((0.26)(1-0.26)/400) = 0.30

4) State your conclusion:

We can be 94% confident that the true proportion of freshmen business majors is between 22% and 30%. 20.4% is not included in the interval.