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Name _______________________________ Period _____________ Test 10 Confidence Intervals Homework (Chpt 10.1, 11.1, 12.1) For 1 & 2, determine the point estimator you would use and calculate its value. 1. How many pairs of shoes, on average, do female teens have? To find out, an AP Statistics class conducted a survey. They selected an SRS of 20 female students from their school. Then they recorded the number of pairs of shoes that each student reported having. Here are the data: 50 26 26 31 57 19 24 22 23 38 13 50 13 34 23 30 49 13 15 51 607 Point estimator would be the sample mean, 𝑥, � 𝑥̅ = = 30.35 20 2. Tonya wants to estimate what proportion of the seniors in her school plan to attend the prom. She interviews an SRS of 50 of the 750 seniors in her school and finds that 36 plan to go to the prom. 36 Point estimator would be the sample proportion, 𝑝̂ , 𝑝̂ = = .72 50 3. Young people have a better chance of full-time employment and good wages if they are good with numbers. One source of data is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey, which is based on a nationwide probability sample of households. The NAEP survey includes a short test of quantitative skills and scores on the test range from 0 to 500. Suppose that you give the NAEP test to an SRS of 840 people from a large population in which the scores have mean 280 and standard deviation 𝜎𝜎 = 60. The mean 𝑥̅ of the 840 scores will vary if you take repeated samples. a) Describe the shape, center, and spread of the sampling distribution of 𝑥̅ . We can’t verify that the population distribution is Normal, but since n = 840 the CLT says the sampling distribution will be approximately Normal. 10% condition: 10(840) = 8400; there are at least 8400 people in the population 60 𝜇𝜇𝑥̅ = 𝜇𝜇 = 280; 𝜎𝜎𝑥̅ = = 2.0702 N(280, 2.0702) √840 b) Sketch the sampling distribution of 𝑥̅ . Mark its mean and the values one, two, and three standard deviations on either side of the mean. 274 276 278 280 282 284 286 c) According to the 68-95-99.7 rule, about 95% of all values of 𝑥̅ lie within a distance m of the mean of the sampling distribution. What is m? Shade the region on the axis of your sketch that is within m of the mean. 2 standard deviations, 𝑚 = 2(2.0702) = 4.1704 d) Whenever 𝑥̅ falls in the region you shaded, the population 𝜇𝜇 lies in the confidence interval 𝑥̅ ± 𝑚. For what percent of all possible samples does this interval capture 𝜇𝜇? 95% of all samples will capture 𝜇𝜇. e) Below your sketch, choose one value of 𝑥̅ inside the shaded region and draw its corresponding confidence interval. Do the same for one value of 𝑥̅ outside the shaded region. What is the most important difference between these intervals? One captures 𝜇𝜇, the other does not. 4. The figure to the right shows the result of taking 25 SRSs from a Normal population and constructing a confidence interval for each sample. Which confidence level – 80%, 90%, 95%, or 99% - do you think was used? Explain. 5. A New York Times/CBS News Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered “Yes”. The article describing the poll says that it “is based on telephone interviews conducted from Sept. 13 to Sept. 18 with 1,664 adults around the United States, excluding Alaska and Hawaii…The telephone numbers were formed by random digits, thus permitting access to both listed and unlisted residential numbers.” The article gives the margin of error for a 95% confidence level as 3 percentage points. a) Explain what the margin of error means to someone who knows little statistics. If we were to repeat the sampling procedure many times, the sample proportion would be within 3 percentage points of the true proportion in 95% of the intervals. b) State and interpret the 95% confidence interval. (.63, .69) We are 95% confident that the interval from .63 to .69 captures the true proportion of those who fav

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