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AP Statistics Section 10.3

AP Statistics Section 10.3.

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AP Statistics Section 10.3

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  1. AP Statistics Section 10.3

  2. Consider the following questions: what proportion of U.S. adults are unemployed right now; what proportion of teenagers have a computer with internet access in their bedrooms; what proportion of college students pray on a daily basis; what proportion of preteens have a cell phone; and what proportion of Californians approve of President Obama’s handling of the situation in Pakistan?

  3. We are interested in the unknown proportion p, of a population that has some outcome. For convenience, call the outcome we are looking for a ___________.

  4. Conditions for Inference about a ProportionReviewing the sampling distribution of a sample proportion from section 9.2 we have:

  5. Center: the mean is ____. That is, the sample proportion is an unbiased estimator of the population proportion p.Spread: The standard deviation of is _____________, provided that the population is at least 10 times as large as the sample.Shape: If the sample size is large enough that both __________ and ______________, then thedistribution of is approximately Normal.

  6. In practice, we don’t know the value of p. So……Shape: In large samples, will be close to p. Therefore, we replace p with in determining the values of np and n(1 – p). Spread: We replace the standard deviation by the __________________ of p. SE =

  7. The conditions for inference about a proportion are:SRS: the data are an SRS from the population of interest.Normality: For a CI, n must be large enough that both _________ and _________________.Independent: Individual observations are independent. When sampling without replacement,the population is at least 10 times as large as the sample.

  8. A Confidence Interval for p: Using z ProceduresDraw an SRS of size n from a large population with unknown proportion p of successes. An approximate level C confidence interval for p iswhere z* is the upper critical value of the standard Normal distribution.

  9. Example: Alcohol abuse is a concern on college campuses. Researchers defined “frequent binge drinking” as having five or more drinks in a row three or more times in the last two weeks. The Harvard School of Public Health sent surveys to a random sample of undergraduate students at 120 colleges and universities. Of the 10,904 students who responded, 2,486 were frequent binge drinkers. Calculate a 99% confidence interval for the proportion p of all college undergraduates who admit to frequent binge drinking. (In order to determine this, we must act as if the sample were an SRS)

  10. Parameter:

  11. ConditionsSRS: Normality: Independence:

  12. Calculations

  13. Interpretation

  14. Note: The margin of error in this confidence interval includes only random sampling error! There are other possible sources of error not taken into account, such as?

  15. Choosing the Sample SizeIn planning a study, we may want to choose a sample size that will allow us to estimate the parameter within a given margin of error. Earlier, we did this for a population mean. The method is similar for estimating a population proportion.

  16. The margin of error in the confidence interval for p is…If we want a given margin error, we consider the inequalityand solve for n.

  17. T o find our necessary sample size, n, we need to know, , the sample proportion of successes. We will need to “guess” this value. Call our guess p*.Here are two ways to get p*.1. Use a guess for p* based on a pilot study or on past experience with similar studies. 2. Use p* = _____ as the guess.

  18. Note: If we hold m and p* constant, using p* =.5, will give us the largest possible value of n. Thus, you are guaranteed of having a large enough sample size. However, sampling costs money, so if you have value for p* other than .5, you would prefer to use it.

  19. Example: A company has received complaints about its customer service. They intend to hire a consultant to carry out a survey of customers. Before contacting the consultant, the company president wants some idea of the sample size that she will be required to pay for. One critical question is the degree of satisfaction with the company’s customer service, measured on a five-point scale. The president wants to estimate the proportion p of customers who are “very satisfied” or “satisfied”. She decides that she wants the estimate to be within 3% and have a 95% confidence level.

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