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**AP Statistics 41 days until the AP Exam**

Sampling Distributions for Proportions Formulas for Proportions Assumption Checking

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Today’s Objective I can apply the properties and formulas of sampling distributions for proportions

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**Quick Review The formula for the mean of a sampling distribution is:**

The formula for the standard deviation of the mean of a sampling distribution is: A parameter is: A statistic is: The symbol for population mean is: The symbol for sample mean is:

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**Sample Proportions Remember that a proportion is simply a fraction.**

The sampling distribution of works in the same way that the sampling distribution of the sample mean does.

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**Formulas for Proportions**

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**Assumptions and Conditions**

The sampled values are independent of each other The sample size n is large enough. Conditions The sample should come from an SRS The sample size should not be larger than 10% of the population The sample size should be sufficiently large so that

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Example 1 A manufacturer of computer printers purchases plastic ink cartridges from a vendor. When a large shipment is received, a random sample of 200 cartridges is selected, and each is inspected. If the sample proportion of defectives is more than .02, the entire shipment will be returned to the vendor. a) What is the approximate probability that the shipment will be returned if the true proportion of defectives in the shipment is .05? Be sure to check the assumptions necessary for accurate probabilities using proportions. b) What is the approximate probability that the shipment will not be returned when the true proportion of defectives in the shipment is .10?

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Example 2 For the years 2000 – 2002, the proportion of mothers in the state of Texas under the age of 18 that gave birth to children less than 2500 grams was 9.6%. a) Draw the sampling distribution of based on a random sample of 200. b) What is the probability that more than 12% of the sample of mothers gave birth to children less than 2500 grams? c) What is the probability that less than 5% of the sample of mothers gave birth to children less than 2500 grams? d) Suppose that the sample for part c. was collecting in Harris County. Comment on how this affect our answer.

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Example 3 The article “Should Pregnant Women Move? Linking Risks for Birth Defects with Proximity to Toxic Waste Sites” (Chance, 1992) reported that in a large study carried out in the state of New York, approximately 30% of the study subjects lived within 1 mile of a hazardous waste site. Let p denote the proportion of all New York residents who live within 1 mile of such a site. a) Draw the sampling distribution of based on a random sample of 400. b) When n = 400, what is ? c) Is the probability in part (b.) larger or smaller than would be the case if n = 500? Think, don’t calculate.

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Example 4 A certain chromosome defect occurs in only one out of 200 Caucasian adult males. a) What is the smallest value of n for which the sampling distribution of is approximately normal? b) For a random sample of 100 adult Caucasian males, what is the probability that the sample proportion is greater than 1%? c) Is the probability in part (b.) larger or smaller than would be the case if n = 200? Think, don’t calculate.

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+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 6: Random Variables Section 6.3 Binomial and Geometric Random Variables.

+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 6: Random Variables Section 6.3 Binomial and Geometric Random Variables.

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