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4.3 Sample Proportions
Definition. A population proportion is the proportion of individuals in a population sharing a certain trait, denoted p. The sample proportion is the proportion of individuals in a sample sharing a certain trait, denoted ˆp.
The Sampling Distribution of ˆp
Note. How good is the statistic ˆp as an estimate of the parameter p? To find out, we ask, “What would happen if we took many samples?” The sampling distribution of ˆp answers this question. In the simulation examples in Section 4.1, we found:
- The sampling distribution of the sample proportion ˆp has a shape that is close to normal.
- Its mean is close to the population proportion p.
- Its standard deviation gets smaller as the size of the sample gets larger.
Definition. Choose an SRS of size n from a large population with population proportion p having some characteristic of interest. Let p be the proportion of the sample having that characteristic. Then:
- The sampling distribution of ˆp is approximately normal and is closer to a normal distribution when the sample size n is large.
- The mean of the sampling distribution is exactly p.
- The standard deviation of the sampling distribution is √√√ √ p(1^ −^ p) n.
Note. As a rule of thumb, use the recipe for the standard deviation of pˆ only when the population is at least 10 times as large as the sample.
Example 4.14. You ask an SRS of 1500 first-year college students whether they applied for admission to any other college. There are over 1.7 million first-year college students, so the rule of thumb is eas- ily satisfied. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that your sample will give a result within 2 percentage points of this true value? We have an SRS of n = 1500 drawn from a population in which the proportion p = .35 applied to other colleges. The sample proportion ˆp has mean 0.35 and standard deviation √ √√ √ p(1^ −^ p) n =
√√√ √ (.35)(.65) 1500 =^.^0123. We want the probability that ˆp falls between 0.33 and 0.37 (within 2 percentage points, or 0.02, of 0.35). This is a normal distribution calculation. Standardize ˆp by subtracting its mean 0.35 and dividing by its standard deviation 0.123. That produces a new statistic that has the standard normal distribution. It is usual to call such a statistic Z:
Z = pˆ.^0123 −^.^35.
Then draw a picture of the areas under the standard normal curve
approximation will be quite accurate. The mean of ˆp is p = .11. The standard deviation is √√√ √ p(1^ −^ p) n =
√√√ √ (.11)(.89) 1500 =^.^00808.
Now do the normal probability calculation illustrated in Figure 4. (and TM-71):
P (ˆp ≤ .092) = P
( (^) pˆ −. 11
. 00808 ≤^
) = P (Z ≤ − 2 .23) =. 0129.
Only 1.29% of all samples would have so few blacks. Because it is unlikely that a sample would include so few blacks, we have good reason to suspect that the sampling procedure underrepresents blacks.
Sample Counts
Note. Sometimes we are interested in the count of special individuals in a sample rather than the proportion of such individuals. To deal with these problems, just restate them in term of proportions.
