Sampling Distribution of a Sample Proportion Lecture 26 Sections 8.1 – 8.2 Fri, Oct 21, 2005

Parameters and Statistics THE PURPOSE OF A STATISTIC IS TO ESTIMATE A POPULATION PARAMETER. A sample mean is used to estimate the population mean. A sample proportion is used to estimate the population proportion. Sample statistics, by their very nature, are variable. Population parameters are fixed.

Example Example 8.1, p. 464. The Census Bureau surveys 3000 employees and asks them, “Have the job skills demanded by your job increased over the past few years?” 57% replied, “Yes.” That is a sample proportion. What is the population proportion?

Some Questions Do you think that the answer will always be the same, namely 57%? Would a sample proportion of 0 be possible? Would a sample proportion of 1 be possible? Do you think any of the resulting sample proportions could be more than 0.75? Do you know if this sample proportion of 0.57 is close to the true population proportion? Why do you think conclusions were drawn from this value?

Some Questions We hope that the sample proportion is close to the population proportion. How close can we expect it to be? Would it be worth it to collect a larger sample? If the sample were larger, would we expect the sample proportion to be closer to the population proportion? How much closer?

The Sampling Distribution of a Statistic Sampling Distribution of a Statistic – The distribution of values of the statistic over all possible samples of size n from that population.

The Sample Proportion Let p be the population proportion. Then p is a fixed value (for a given population). Let p^ (“p-hat”) be the sample proportion. Then p^ is a random variable; it takes on a new value every time a sample is collected. The sampling distribution of p^ is the probability distribution of all the possible values of p^.

Example Suppose that this class is 3/4 freshmen. Suppose that we take a sample of 2 students, selected with replacement. Find the sampling distribution of p^.

Example P(FF) = 9/16 F P(FN) = 3/16 P(NF) = 3/16 N P(NN) = 1/16 3/4 1/4 P(FF) = 9/16 P(FN) = 3/16 P(NF) = 3/16 P(NN) = 1/16

Example Let X be the number of freshmen in the sample. The probability distribution of X is x P(X = x) 1/16 1 6/16 2 9/16

Example Let p^ be the proportion of freshmen in the sample. (p^ = X/n.) The sampling distribution of p^ is x P(p^ = x) 1/16 1/2 6/16 1 9/16

Samples of Size n = 3 If we sample 3 people (with replacement) from a population that is 3/4 freshmen, then the proportion of freshmen in the sample has the following distribution. x P(p^ = x) 1/64 = .02 1/3 9/64 = .14 2/3 27/64 = .42 1

Samples of Size n = 4 If we sample 4 people (with replacement) from a population that is 3/4 freshmen, then the proportion of freshmen in the sample has the following distribution. x P(p^ = x) 1/256 = .004 1/4 12/256 = .05 2/4 54/256 = .21 3/4 108/256 = .42 1 81/256 = .32

Sampling Distributions Run the program Central Limit Theorem for Proportions.exe. Use n = 30 and p = 0.75; generate 100 samples.

100 Samples of Size n = 30  = 0.75  = 0.079

Observations and Conclusions Observation #1: The values of p^ are clustered around p. Conclusion #1: p^ is probably close to p.

Central Limit Theorem for Proportions.exe. Larger Sample Size Now we will select 100 samples of size 120 instead of size 30. Run the program Central Limit Theorem for Proportions.exe. Pay attention to the spread (standard deviation) of the distribution.

100 Samples of Size n = 120  = 0.75  = 0.0395

Observations and Conclusions Observation #2: As the sample size increases, the clustering is tighter. Conclusion #2A: Larger samples give more reliable estimates. Conclusion #2B: For sample sizes that are large enough, we can make very good estimates of the value of p.

Central Limit Theorem for Proportions.exe. Larger Sample Size Now we will select 10000 samples of size 120 instead of only 100 samples. Run the program Central Limit Theorem for Proportions.exe. Pay attention to the shape of the distribution.

10,000 Samples of Size n = 120  = 0.75  = 0.0395

10,000 Samples of Size n = 126

More Observations and Conclusions Observation #3: The distribution of p^ appears to be approximately normal.

One More Conclusion Conclusion #3: We can use the normal distribution to calculate just how close to p we can expect p^ to be. However, we must know the values of  and  for the distribution of p^. That is, we have to quantify the sampling distribution of p^.

The Sampling Distribution of p^ It turns out that the sampling distribution of p^ is approximately normal with the following parameters. This is the Central Limit Theorem for Proportions, summarized on page 519.

The Sampling Distribution of p^ The approximation to the normal distribution is excellent if

Example Suppose 51% of the population plan to vote for candidate X, i.e., p = 0.51. What is the probability that an exit survey of 1000 people would show candidate X with less than 45% support, i.e., p^ < .45?

Example First, describe the sampling distribution of p^ if the sample size is n = 1000 and p = 0.51. Check: np = 510  5 and n(1 – p) = 490  5. p^ is approximately normal. p^ = 0.51. p^ = ((.51)(.49)/1000) = 0.01581.

Example The z-score of 0.45 is z = (0.45 – 0.51)/.01581 = -3.795. P(p^ < 0.45) = P(Z < -3.795) = 0.00007385 (not likely!) That is why surveys work (within the margin of error).

Let’s Do It! Let’s do it! 8.5, p. 521 – Probabilities about the Proportion of People with Type B Blood. Let’s do it! 8.6, p. 523 – Estimating the Proportion of Patients with Side Effects. Let’s do it! 8.7, p. 525 – Testing hypotheses about Smoking Habits.