Simulating a Sample Distribution The height of young women is approximately normal with a mean of 64.5 inches and a standard deviation of 2.5. 1) Use your calculator to randomly select 100 women. I will show you how… 2) Create a histogram of the 100 heights. Draw this in your notes. 3) Find the average height for your sample; round to the nearest tenth. 4) Do steps 1 and 3 (NOT 2) two more times so that you have 3 total averages. 5) Write your averages on sticky notes and put them on the board to create a histogram for the average heights. Scale axes on board by 0.2 from 62-68

Section 7.2 Sample Proportions

Where We’ve Been… 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 the sampling distribution of . We will calculate the mean and standard deviation of the sampling distribution of . Can you find the formulas????

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.

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

Rules of Thumb and the Normal Approximation We can use the normal approximation for p-hat ONLY when np ≥ 10 AND n(1-p) ≥ 10. *Notice no hat on p!! 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.

Example A polling organization asks a 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?

Key Points: All of these must be covered in your answer to get full credit!! Always define the population of interest AND your RV. State the values of n, p, and 1-p. Check BOTH rules by plugging in values. Convert to a Z-score. Make sure you calculate the mean and standard deviation for the problem. State the probability with symbols. Find the probability using Table A. Write your conclusions in context of the problem.

Next Example One way of checking under-coverage and non-response is to compare the sample with known facts about the population. Suppose 11% of Americans are Asian. The proportion p-hat of Asians in an SRS of 1500 adults, therefore, should be close to 0.11. If a national survey contains only 9.2% Asians, should we be suspect that the sampling procedure is somehow biased, under-representing the Asian population? To answer, we will find the probability that a sample of size 1500 contains no more than 9.2% Asians.

Determine the sample size needed for a set standard deviation: Plug into the formula: Solve for n. **If they don’t give you p, use 0.5 as an estimate. Determine the sample size needed to have a standard deviation of 0.013 If p = 0.35 If p is unknown