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Sampling Distribution of a Sample Proportion Lecture 25 Sections 8.1 – 8.2 Fri, Feb 29, 2008

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Sampling Distributions Sampling Distribution of a Statistic

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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 ^.

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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 ^.

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

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Example Let x be the sample number of males who wash. The probability distribution of x is xP(x)P(x) 01/3 12/3

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Example Let p ^ be the sample proportion of males who wash. (p ^ = x/n.) The sampling distribution of p ^ is p^p^ P(p^)P(p^) 01/3 12/3

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

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

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Example Let x be the sample number of males who wash. The probability distribution of x is xP(x)P(x) 01/9 14/9 2

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Example Let p ^ be the sample proportion of males who wash. (p ^ = x/n.) The sampling distribution of p ^ is p^p^ P(p^)P(p^) 01/9 1/24/9 1

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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(p^)P(p^) 01/27 =.03 1/36/27 =.22 2/312/27 =.44 18/27 =.30

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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(p^)P(p^) 01/81 =.01 1/48/81 =.10 2/424/81 =.30 3/432/81 =.40 116/81 =.20

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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(p^)P(p^) 01/243 =.004 1/510/243 =.041 2/540/243 =.165 3/580/243 =.329 4/580/243 =.329 132/243 =.132

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Our Experiment In our experiment, we had 80 samples of size 5. Based on the sampling distribution when n = 5, we would expect the following Value of p ^ 0.00.20.40.60.81.0 Actual Predicted0.33.313.226.3 10.5

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The pdf when n = 1 01

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The pdf when n = 2 011/2

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The pdf when n = 3 011/32/3

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

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The pdf when n = 5 011/52/53/5 4/5

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1 8/10 The pdf when n = 10 02/104/106/10

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

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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.

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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.

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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 ^.

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

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

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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.

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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

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

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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%?

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Why Surveys Work Now we have

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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%.

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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?

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Quality Control A company will accept a shipment of components if there is no strong evidence that 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.

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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 ?

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