 A national opinion poll recently estimated that 44% (p-hat =.44) of all adults agree that parents of school-age children should be given vouchers good for education at any public or private school of their choice. The polling organization used a probability sampling method for which the sample proportion p-hat has a normal distribution with standard deviation about If a sample of the same size were drawn by the same method from the state of New Jersey (population 7.8 million) instead of from the entire United States (population 280 million), would this standard deviation be larger, about the same, or smaller? Explain your answer.

Section 7.2 Sample Proportions

Where We’ve Been…  Let’s think about what we’ve learned so far in this chapter. We’ve distinguished between statistics and parameters and used the appropriate symbols for each. We’ve learned what a sampling distribution is, how to simulate creating a sampling distribution, and how to graph and describe a sampling distribution. We’ve learned the difference between bias and variability of a statistic.

What’s in Store…  Today, we’ll focus on one sampling distribution – the sampling distribution of.  So, we’re going to talk about the mean and standard deviation of the sampling distribution of.

The Sampling Distribution of P-hat In words, the mean of the sampling distribution of p-hat is p. That makes p-hat an unbiased estimator of p. Let’s find these formulas on our formula sheet. Remember – these are NOT binomial, they are for sampling distributions of p-hat.

Sample size and σ  What happens to σ when n is large?  Therefore, as our sample size gets larger, it has less variability.

Rules to live by  Also, we learned that a population should be at least 10 times the size of the sample.  We learned that a sampling distribution is approximately normal, if the sample size is large.

They apply to sample proportion distributions as well  We can use the normal approximation for p-hat ONLY when np ≥ 10 AND n(1-p) ≥ 10.  We can use the formula for the standard deviation of p-hat only when the population is at least 10 times the sample size. In symbols, population ≥ 10n.

Key Points  State the values of n, p, and 1-p.  Check BOTH rules of thumb by plugging in values. SHOW THIS!!!  Graph the distribution you’re interested in.  Convert to a Z-score. Make sure you know what the mean and standard deviation are for the problem.  State the probability with symbols. Find the probability using Table A.  Write your conclusions in words in the context of the problem.

Example  A polling organization asks an SRS of 1500 first-year 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 the true value?

Next Example  One way of checking under coverage and non-response is to compare the sample with known facts about the population. Suppose 5.6% of Americans are Asian. The proportion p-hat of Asians in an SRS of 1500 adults, therefore, should be close to If a national survey contains only 3.8% Asians, should we be suspect that the sampling procedure is somehow under representing Asians? To answer this, we will find the probability that a sample of size 1500 contains no more than 3.8% Asians.

A little tricky  So when do I just use a normal curve without special rules? You are only looking at ONE individual.  When do I use my new rules? You are looking at a SAMPLE.

CHECK YOUR UNDERSTANDING P. 437

Homework p. 439 (28, 30, 36, 38, 40)