People end up tossing 12 of what they buy at the grocery...

People end up tossing 12$\%$ of what they buy at the grocery store (Reader's Digest, March, 2009 . Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior. a. Show the sampling distribution of $\overline{p},$ the proportion of grocenies thrown out by your sample respondents. b. What is the probability that your survey will provide a sample proportion within $\pm .03$ of the population proportion? c. What is the probability that your survey will provide a sample proportion within $\pm .015$ of the population proportion?

Transcript

00:04 Suppose that people throw out 12 % of what they end up buying at the grocery store.

00:11 We are conducting a sample, a survey, consisting of 540 people, and we want to look at what proportion is thrown out by our sample, by our 540 people.

00:29 So we want to look at the sampling distribution of p -hat, of the sample.

00:35 Portions.

00:37 We know that 12 % of all people, the proportion of the population, 12 % is thrown out.

00:47 12 % of everything that's purchased at the store is thrown out.

00:51 So p is 0 .12.

00:53 That's the true population proportion.

00:55 But in our sample of 540 people, perhaps our particular sample may be our sample ends up throwing out a little bit more than 12 % of what they buy at the store, maybe a little bit less.

01:12 So we're going to start talking about the sample distribution of the sample proportions.

01:18 P -hat and k represents the little arrow on top of the p -hat is the sample proportion.

01:26 Okay, we want to look at the distribution of the sample proportions.

01:33 Okay, population as a whole throws out 12 % of what they buy.

01:39 We have a sample.

01:41 540 people is our sample size.

01:44 We have a sample and perhaps our sample throws out more than 12 percent.

01:49 Perhaps they throw out less than 12 percent of what they buy.

01:53 The sample distribution of the sample proportions basically shows all the different sample proportions that you can get when you have a sample size of 540.

02:10 A bunch of samples, each one has 540 people in them.

02:16 What proportion is being thrown out by each sample? maybe you have a sample of 540 people that throws out exactly 12 % of what they buy.

02:26 Maybe you have another sample that throws out 13%, 14%.

02:30 Maybe other samples will throw out just 11 % or maybe all the way down.

02:35 Down to just 8%.

02:36 So that's what we mean by the distribution, the sample distribution of the sample proportions.

02:43 Now, the sample distribution of the sample proportions will have a mean.

02:56 The sample distribution of the sample proportions will have a mean, that mean will equal the population proportion.

03:05 0 .12.

03:09 The standard deviation of our sample distribution of sample proportions will be given by this formula.

03:18 The square root of p times 1 minus p over n.

03:29 So the standard deviation of our sample distribution of the sample proportions is going to be the square root of p, which is 0 .12 times 1 minus p, 1 minus 0 .12 is 0 .88 divided by n our sample size.

03:51 So 0 .12 times 0 .88 divided by n our sample size, 540 people in our sample.

04:03 So we have to do the square root of 0 .12 times 0 .88 divided by 540.

04:13 So 0 .12 times .88 divided by 540...

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