Sampling Distribution of a Sample Proportion Lecture 28 Sections 8.1 – 8.2 Wed, Mar 7, 2007

Sampling Distributions Sampling Distribution of a Statistic

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 1 student. Find the sampling distribution of p ^.

Example F N 3/4 1/4 P(F) = 3/4 P(N) = 1/4

Example Let X be the number of freshmen in the sample. The probability distribution of X is xP(x)P(x) 01/4 13/4

Example Let p ^ be the proportion of freshmen in the sample. (p ^ = X/n.) The sampling distribution of p ^ is xP(p ^ = x) 01/4 13/4

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

Example F N F N F N 3/4 1/4 3/4 1/4 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 xP(x)P(x) 01/16 16/16 29/16

Example Let p ^ be the proportion of freshmen in the sample. (p ^ = X/n.) The sampling distribution of p ^ is xP(p ^ = x) 01/16 1/26/16 19/16

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. xP(p ^ = x) 01/64 =.02 1/39/64 =.14 2/327/64 =.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. xP(p ^ = x) 01/256 =.004 1/412/256 =.05 2/454/256 =.21 3/4108/256 = /256 =.32

The pdf when n = 1 01

The pdf when n = 2 011/2

The pdf when n = 3 01

The pdf when n = 4 011/42/43/4

The pdf when n = 8 011/42/43/4

The pdf when n = /42/43/4

The pdf when n = /42/43/4

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

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

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

Why Surveys Work Suppose that we are trying to estimate the proportion of the population who own a cell phone. Suppose the true proportion is 75%. 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 First, describe the sampling distribution of p ^ if the sample size is n = 1000 and p =  Check: np = 750  5 and n(1 – p) = 250  5, so p ^ is approximately normal.

Why Surveys Work Then find the parameters  p^ and  p^.

Why Surveys Work Now find the probability that p^ is between 0.70 and normalcdf(.70,.80,.75,.01369) = It is virtually certain that our estimate with be within 5% of 75%.

Why Surveys Work What if we had surveyed only 200 people?

Surveys What range of percentages contains 95% of the sample proportions?

Surveys Suppose that Candidate X has 48% of the vote and Candidate Y has 52% of the vote. What is the probability that a survey of 100 people will indicate that Candidate X is ahead?

Surveys What is the probability that a survey of 2000 people will indicate that Candidate X is ahead?

Quality Control A company will accept a shipment of components if they are convinced that no more than 5% of them are defective. H 0 : 5% of the parts are defective. H 1 : 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  ?