Now we continue our discussion of identifying unusual sample proportions. The type of thinking we do here prepares us for the type of thinking we will do in statistical inference.

EXAMPLE

According to a report by the Pew Research Center, in October 2007 about 10% of the 3.1 million 18- to 24-year-olds in the United States were enrolled in a community college.

Suppose in that year we randomly selected 100 young adults in this age group and find that 15% of the sample was enrolled in a community college.

Is a result of 0.15 unlikely given that the population proportion is 0.10? Is it outside of the margin of error (outside two standard errors)?

We can answer this question only by looking at the variability in sample proportions for this situation.

We will use an applet to simulate the selecting of random samples. This applet simulates the random selection of 5,000 samples with a sample size n from a population with proportion p. For each sample, the applet plots the sample proportion and shows the resulting histogram of sample proportions.

For this example,

Set p = 0.10 in the applet because the population proportion is 0.10.
Set n=100 to represent the sample size of 100.
Use the slider to set the sample proportion at 0.15. (If you click on the button in the slider, you can use left or right arrows on your keyboard to move the slider in increments of 0.05)
Check the box "Show Standard Deviation Bars"
Hit the “Run simulation (5,000 samples)” button.

Run the simulation using the applet below or open the applet in a separate tab.

Below is a picture of our results. Your results will be close to these.

$Normal distribution with p-hat marked at 0.15 and 1SD and 2SD bars$

Analysis:

The mean of the sample proportions is p=0.10 (or something very close; we got 0.0999 in our simulation.)
The standard deviation of the sample proportions is 0.03 (or something very close.) The standard deviation describes the average amount of error in sample proportions that is due to chance. On average, sample proportions will have a 3% error.
A sample proportion of 0.15 has a 5% error (0.15 – 0.10 = 0.05), so this is a larger error than we expect on average.
However, the sample proportion of 0.15 is within the margin of error of 6% (0.03 x 2 = 0.06). In other words, 0.15 is within 2 standard errors of the population proportion 0.10. Therefore, 0.15 is not considered a highly unlikely result.

Conclusion:

For random samples of 100 young adults, a sample with 15% enrolled in a community college is unusual if only 10% of the population overall is enrolled but it is not highly unlikely. $End of example$

EXAMPLE

Here is another way to look at it. We marked off the intervals of sample proportions within one and two standard deviations of the mean.

$applet image - distribution of sample proportions with p = 0.1 and n = 100$

Typical samples have sample proportions within 1 standard error of 0.10: between 0.07 and 0.13 (just subtract 0.03 from 0.10 and then add 0.03 to 0.10).
Most sample proportions fall within about 2 standard errors of 0.10: between 0.04 and 0.16. (just subtract 0.06 from 0.10 and then add 0.06 to 0.10). It is extremely unusual for sample proportions to have values outside of this range.
Therefore, when sampling from a population with p=0.10, a sample proportion of 0.15 is not typical (falls outside of the interval 0.07 to 0.13), but it is also not extremely unusual (falls within the interval 0.04 to 0.16). $End of example$

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