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S AMPLE P ROPORTIONS

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W HAT DO YOU THINK ? Are these parameters or statistics? What specific type of parameter/statistic are they? How do you think they were calculated? How can we be sure that method yielded a good estimate? 91% of teens have been bullied 74% of US teens carry a cell phone 57% of teens credit their cell phone with improving their life 42% of teens can text blindfolded Source: stageoflife.com

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S AMPLE P ROPORTIONS To figure out how we determine a proportion of interest in a large population. So we can gain information about populations even when we can’t survey the entire population. ObjectivePurpose

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION The sample proportion is the statistic that we use to gain information about the unknown population parameter p. How good is the statistic as an estimate of the parameter p ? To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION The sample proportion is the statistic that we use to gain information about the unknown population parameter p. How good is the statistic as an estimate of the parameter p ? To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION The sample proportion is the statistic that we use to gain information about the unknown population parameter p. How good is the statistic as an estimate of the parameter p ? To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION The sample proportion is the statistic that we use to gain information about the unknown population parameter p. How good is the statistic as an estimate of the parameter p ? To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION The sample proportion is the statistic that we use to gain information about the unknown population parameter p. How good is the statistic as an estimate of the parameter p ? To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION Standard Deviation ( ) Mean ( ) Values of

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION Standard Deviation ( ) Mean ( ) Values of

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION Standard Deviation ( ) Mean ( ) Values of

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION Chapter 7: If then Chapter 8: = count of “successes” in sample size of sample in ghosts. ToolboxMean Ex: Do You Believe in Ghosts? 160 / 515 = 0.31 said yes!

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S AMPLING D ISTRIBUTION OF A S AMPLE P ROPORTION ToolboxStandard Deviation Chapter 7: If then Chapter 8:

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I N E NGLISH, P LEASE ! The mean of the sampling distribution of a sample proportion is exactly p. The standard deviation of the sampling distribution of a sample proportion is

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S O W HAT ? The sample proportion is an unbiased estimator of p ! The standard deviation of gets smaller as the sample size n increases. That is, is less variable in larger samples.

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S O W HAT ? The sample proportion is an unbiased estimator of p ! The standard deviation of gets smaller as the sample size n increases. That is, is less variable in larger samples.

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S O W HAT ? The sample proportion is an unbiased estimator of p ! The standard deviation of gets smaller as the sample size n increases. That is, is less variable in larger samples.

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S O W HAT ? The sample proportion is an unbiased estimator of p ! The standard deviation of gets smaller as the sample size n increases. That is, is less variable in larger samples.

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W ARNING ! The formula for the standard deviation of doesn’t apply when the sample is a large part of the population. (In that case we could just examine the entire population!) Rule of Thumb 1 : Use the formula for the standard deviation of only when the population is at least 10 times as large as the sample.

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U SING THE N ORMAL A PPROXIMATION The sampling distribution of is approximately normal The larger the sample size n, the closer the sampling distribution is to a normal distribution Rule of Thumb 2: We will use the normal approximation to the sampling distribution of for values of n and p that satisfy np ≥ 10 and n(1-p) ≥ 10.

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E XAMPLE : D O Y OU J OG ? (H OMEWORK P ROBLEM #25) The Gallup Poll once asked a random sample of 1540 adults, “Do you happen to jog?” Suppose that in fact 15% of all adults jog. a) Find the mean and standard deviation of the proportion of the sample who jog. (Assume the sample is an SRS.) b) Explain why you can use the formula for the standard deviation of in this setting. c) Check that you can use the normal approximation for the distribution of. d) Find the probability that between 13% and 17% of the sample jog. e) What sample size would be required to reduce the standard deviation of the sample proportion to one-third the value you found in (a)?

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E XAMPLE : R ULES OF T HUMB (H OMEWORK P ROBLEM #30) Explain why you cannot use the methods of this section to find the following probabilities. a) A factory employs 3000 unionized workers, of whom 30% are Hispanic. The 15-member union executive committee contains 3 Hispanics. What would be the probability of 3 or fewer Hispanics if the executive committee were chosen at random from all the workers? b) A university is concerned about the academic standing of its intercollegiate athletes. A study committee chooses an SRS of 50 of the 316 athletes to interview in detail. Suppose that in fact 40% of the athletes have been told by coaches to neglect their studies on at least one occasion. What is the probability that at least 15 in the sample are among this group? c) Use what you learned in Chapter 8 to find the probability described in part (a).

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R EESE ’ S P IECES A CTIVITY ! Learning Goal: To understand the effect of sample size on the sampling distribution. www.Rossmanchance.com/applets/Reeses/Reeses Pieces.html

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C LOSING S UMMARY Today we continued learning about sampling distributions. We looked at one particular type of sampling distribution – the sampling distribution of the sampling proportion. We learned that as long as the population (Rule of Thumb 1) and sample size (Rule of Thumb 2) are large enough, the sampling distribution is approximately normal with mean p and standard deviation

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E XIT T ICKET Suppose I want to know what proportion of U.S. teenagers play a sport. How could I come up with an answer to my question? Why does that method work? How could I improve my results?

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E XAMPLE : A PPLYING TO C OLLEGE A polling organization asks an SRS of 1500 first- year college students whether they applied for admission to any other college. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value? What are we given? What are we looking for? What does our sampling distribution look like? Is our population at least 10 times the size of our sample? Can we use the normal approximation?

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+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 9: Sampling Distributions Section 9.2 Sample Proportions.

+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 9: Sampling Distributions Section 9.2 Sample Proportions.

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