AP Statistics Section 9.2 Sample Proportions

  The objective of some statistical applications is to reach a conclusion about a population proportion, p, by using the sample proportion, . For example, we may try to estimate an approval rating through a survey or test a claim about the proportion of defective light bulbs in a shipment based on a random sample. Since p is unknown to us, we must base our conclusion on a sample proportion, .

However, as we have seen, the value of will vary from sample to sample However, as we have seen, the value of will vary from sample to sample. The amount of variability will depend upon the ________________

For example: A polling organization asks an SRS of 1500 college students whether they applied for admission to any other college. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value?

Before we can answer this question, we need to take a closer look at the center, shape and spread of the sampling distribution for .

Take a SRS from the population of interest.

Since values of X and will vary in repeated samples, both X and are random variables. Provided the population is at least 10 times the sample size, the count X will follow a binomial distribution. So, ____ and __________.

Now , , so use the transformation rules: If Y = a + bX, then

Rule of Thumb 1 This formula for the standard deviation of can only be used when the population is at least 10 times as large as the sample.

We saw with our simulations in Section 9 We saw with our simulations in Section 9.1, that our sampling distribution of gets closer and closer to a Normal distribution when the sample size, n, is large.

Rule of Thumb 2: Use the Normal approximation to the sampling distribution of for values of n and p that satisfy ________ and ______________. Note that these are the same conditions necessary to use a Normal distribution to approximate a Binomial distribution.

Summarizing the Sampling Distribution for Proportions

If we take repeated random samples of size n from a population, the sample proportion , will have the following distribution and properties.

A polling organization asks an SRS of 1500 college students whether they applied for admission to any other college. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value?

Example: Based on Census data, we know 11% of US adults are black Example: Based on Census data, we know 11% of US adults are black. Therefore p = 0.11. We would expect an SRS to have roughly an 11% black representation. Suppose a sample of 1500 adults contains 138 black individuals. We would not expect to be exactly 0.11 because of sampling variability, but, is this number lower than what would be expected by chance (i.e. should we suspect “undercoverage” in the sample method)?