Example

Community College Students and Gender

According to a 2010 report from the American Council on Education, females make up 57% of the college population in the United States. Students in a statistics class at Tallahassee Community College want to determine the proportion of female students at TCC. They select a random sample of 135 TCC students and find that 72 are female, which is a sample proportion of 72 / 135 ≈ 0.533. So 53.3% of the students in the sample are female.

What can they conclude about the proportion of females at the college? How confident can they be in their estimate?

To answer these questions, we need to find a confidence interval.

Checking conditions:

We learned in Linking Probability to Statistical Inference that a confidence interval comes from a normal model of the sampling distribution. Let’s first make sure that a normal model is appropriate here. Recall the two conditions for using a normal model for sample proportions:

• The sample must be random.
• The expected number of successes in the sample, np, and the expected number of failures, n(1 – p), are both greater than or equal to 10. In symbols, this is np ≥ 10 and n(1 − p) ≥ 10. Recall that success doesn’t mean good and failure doesn’t mean bad. A success is just what we are counting.

When we try to check these conditions, we have a problem. We do not know p, the population proportion. In fact, p is what we are trying to estimate! So we cannot determine the expected number of successes and failures. Our solution to this problem is to adjust these conditions. Advanced theory tells us that if the actual number of successes and failures in the sample are greater than or equal to 10, then a normal model is still a good fit.

This sample contains 72 successes (female students) and 63 failures (male students). Both are greater than 10. We therefore use the normal model for the sampling distribution.

Finding the margin of error:

We know that a sample proportion is only an estimate for the population proportion. We do not expect the sample proportion to equal the population proportion, so there is some error due to random chance. We use the standard deviation of the sample proportions to describe the amount of error we can expect in random samples. We call this the standard error.

In Linking Probability to Statistical Inference, we learned that the standard error of the sample proportion depends on the population proportion and sample size. Here is the formula for the standard error:

[latex]\sqrt{\frac{p(1-p)}{n}}[/latex]

When we use a normal model for the sampling distribution, 95% of sample proportions estimate the population proportion within approximately 2 standard errors. So the margin of error is the following:

[latex]2\text{}\sqrt{\frac{p(1-p)}{n}}[/latex]

Now let’s calculate the margin of error for the TCC estimate of 53.3%. Notice that we have the same problem we had earlier. We don’t know p, the population proportion. So we can’t calculate the margin of error! Our solution to this problem is to estimate the standard error using the sample proportion in place of p. We call this the estimated standard error, and the formula is:

[latex]\sqrt{\frac{\stackrel{ˆ}{p}(1-\stackrel{ˆ}{p})}{n}}[/latex]

For this example, the estimated standard error is

[latex]\sqrt{\frac{0.533(1-0.533)}{135}}\text{}\approx \text{}0.043[/latex]

So the margin of error for the 95% confidence interval is:

[latex]2\text{}\sqrt{\frac{0.533(1-0.533)}{135}}\text{}\approx \text{}2(0.043)\text{}=\text{}0.086[/latex]

Finding the confidence interval:

We can interpret the margin of error by saying we are 95% confident that the proportion of all students at TCC who are female is within 0.086 of our sample proportion of 0.533. We can then write the interval in the following form:

[latex]\stackrel{ˆ}{p}\text{}±\text{}\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error}=0.533\text{}±\text{}0.086[/latex]

When we add and subtract the margin of error from the sample proportion, the confidence interval is 0.447 to 0.619.

Conclusion:

We are 95% confident that the proportion of all TCC students who are female is between 0.447 and 0.619. We can also make this statement using percentages. We are 95% confident that the percentage of all TCC students who are female is between 44.7% and 61.9%.

Recall from Linking Probability to Statistical Inference that 95% confidence means this method captures the population proportion about 95% of the time.