Sampling Distribution of a Sample Proportion Lecture 28 Sections 8.1 – 8.2 Mon, Oct 15, 2007

Sampling Distributions Sampling Distribution of a Statistic

The Sample Proportion The letter p represents the population proportion. The symbol p^ (“p-hat”) represents the sample proportion. p^ is a random variable. The sampling distribution of p^ is the probability distribution of all the possible values of p^.

Example Suppose that 2/3 of all males wash their hands after using a public restroom. Suppose that we take a sample of 1 male. Find the sampling distribution of p^.

Example W N 2/3 1/3 P(W) = 2/3 P(N) = 1/3

Example Let x be the sample number of males who wash. The probability distribution of x is x P(x) 1/3 1 2/3

Example Let p^ be the sample proportion of males who wash. (p^ = x/n.) The sampling distribution of p^ is p^ P(p^) 1/3 1 2/3

Example Now we take a sample of 2 males, sampling with replacement. Find the sampling distribution of p^.

Example W N 2/3 1/3 P(WW) = 4/9 P(WN) = 2/9 P(NW) = 2/9 P(NN) = 1/9

Example Let x be the sample number of males who wash. The probability distribution of x is x P(x) 1/9 1 4/9 2

Example Let p^ be the sample proportion of males who wash. (p^ = x/n.) The sampling distribution of p^ is p^ P(p^) 1/9 1/2 4/9 1

Samples of Size n = 3 If we sample 3 males, then the sample proportion of males who wash has the following distribution. p^ P(p^) 1/27 = .03 1/3 6/27 = .22 2/3 12/27 = .44 1 8/27 = .30

Samples of Size n = 4 If we sample 4 males, then the sample proportion of males who wash has the following distribution. p^ P(p^) 1/81 = .01 1/4 8/81 = .10 2/4 24/81 = .30 3/4 32/81 = .40 1 16/81 = .20

Samples of Size n = 5 If we sample 5 males, then the sample proportion of males who wash has the following distribution. p^ P(p^) 1/243 = .004 1/5 10/243 = .041 2/5 40/243 = .165 3/5 80/243 = .329 4/5 1 32/243 = .132

Our Experiment In our experiment, we had 60 samples of size 5. Based on the sampling distribution when n = 5, we would expect the following Value of p^ 0.0 0.2 0.4 0.6 0.8 1.0 Actual 1 3 7 18 21 10 Predicted 2.5 9.9 19.8 7.9

The pdf when n = 1 1

The pdf when n = 2 1/2 1

The pdf when n = 3 1/3 2/3 1

The pdf when n = 4 1/4 2/4 3/4 1

The pdf when n = 5 1/5 2/5 3/5 4/5 1

The pdf when n = 10 2/10 4/10 6/10 8/10 1

Observations and Conclusions Observation: The values of p^ are clustered around p. Conclusion: p^ is close to p most of the time.

Observations and Conclusions Observation: As the sample size increases, the clustering becomes tighter. Conclusion: Larger samples give better estimates. Conclusion: We can make the estimates of p as good as we want, provided we make the sample size large enough.

Observations and Conclusions Observation: The distribution of p^ appears to be approximately normal. Conclusion: We can use the normal distribution to calculate just how close to p we can expect p^ to be.

One More Observation 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 Central Limit Theorem for Proportions It turns out that the sampling distribution of p^ is approximately normal with the following parameters.

The Central Limit Theorem for Proportions The approximation to the normal distribution is excellent if

Example If we gather a sample of 100 males, how likely is it that between 60 and 70 of them, inclusive, wash their hands after using a public restroom? This is the same as asking the likelihood that 0.60  p^  0.70.

Example Use p = 0.66. Check that np = 100(0.66) = 66 > 5, n(1 – p) = 100(0.34) = 34 > 5. Then p^ has a normal distribution with

Example So P(0.60  p^  0.70) = normalcdf(.60,.70,.66,.04737) = 0.6981.

Why Surveys Work Suppose that we are trying to estimate the proportion of the male population who wash their hands after using a public restroom. Suppose the true proportion is 66%. If we survey a random sample of 1000 people, how likely is it that our error will be no greater than 5%?

Why Surveys Work Now we have

Why Surveys Work Now find the probability that p^ is between 0.61 and 0.71: normalcdf(.61, .71, .66, .01498) = 0.9992. It is virtually certain that our estimate will be within 5% of 66%.

Why Surveys Work What if we had decided to save money and surveyed only 100 people? If it is important to be within 5% of the correct value, is it worth it to survey 1000 people instead of only 100 people?

Quality Control A company will accept a shipment of components if there is no strong evidence that more than 5% of them are defective. H0: 5% of the parts are defective. H1: More than 5% of the parts are defective.

Quality Control They will take a random sample of 100 parts and test them. If no more than 10 of them are defective, they will accept the shipment. What is ? What is ?