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Putting it Together: The Four Step Process 8.2b h.w: pg 497: 39, 43, 47 Target Goal: I can carry out the steps for constructing a confidence interval. I can determine the sample size required to obtain a level C confidence interval.

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Review: Finding a Critical ValueUse Table A to find the critical value z* for an 80% confidence interval. Assume that the Normal condition ismet. Estimating a Population Proportion Since we want to capture the central 80% of the standard Normal distribution, we leave out 20%, or 10% in each tail. Search Table A to find the point z* with area 0.1 to its left. So, the critical value z* for an 80% confidence interval is z* = 1.28. Try invnorm(990) The closest entry is z = – 1.28. z.07.08.09 – 1.3.0853.0838.0823 – 1.2.1020.1003.0985 – 1.1.1210.1190.1170

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The Four-Step ProcessWe can use the familiar four-step process whenever a problem asks us to constructand interpret a confidence interval. Estimating a Population Proportion State: What parameter do you want to estimate, and at what confidence level? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, perform calculations. Conclude: Interpret your interval in the context of the problem. State: What parameter do you want to estimate, and at what confidence level? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, perform calculations. Conclude: Interpret your interval in the context of the problem. Confidence Intervals: A Four-Step Process

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Ex. Binge Drinking in College In a representative of 140 colleges and 17592 students, 7741 students identify themselves as binge drinkers. Considering this SRS, construct a 95% confidence interval for the proportion of students who identify themselves as binge drinkers.

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Step 1: Identify the population of interest and the parameter you want to draw a conclusion about. State: We want to estimate the actual proportion of all college students who identify themselves as binge drinkers at a 95% confidence level.

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Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure. Plan: We will use a one-sample z interval for p if the conditions are met. Random: SRS? Yes given. Independent: Total population > 10 n: 10(17592) = there are more than 175,920 college students in the country ( to use sample σ) so yes Normal: 17592(.44) = 7740 ≥ 10 17592(.56) = 9851 ≥ 10 Yes, we can use normal approximation.

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Step 3: DO - If the conditions are met, perform calculations. Diagram: invnorm(1-.025) z* = (table A or calc.)

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Ex. Estimating Risky Behavior

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Step 4: Conclude We are 95% confident that the actual percent of all college students who identify themselves as binge drinkers lies between 43% and 45%.

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Ex. Is that Coin Fair? The French naturalist Count Buffon tossed a coin 4040 times and counted 2048 heads. The sample proportion of heads is = 0.5069

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Ex. Confidence Interval for p Calculate the 95% C.I. for the probability p that Buffon’s coin gives a head. (Do and Conclude only)

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= (.4915, 0.5223) We are 95% confident that the probability of a tossing ahead is between 0.4915 and 0.5223.

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Now try: STAT: TESTS:1-Prop Z Int

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Choosing the Sample SizeIn planning a study, we may want to choose a sample size that allows us to estimate a population proportion within a given margin of error. Estimating a Population Proportion The margin of error (ME) in the confidence interval for p is z* is the standard Normal critical value for the level of confidence we want. To determine the sample size n that will yield a level C confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n: Sample Size for Desired Margin of Error

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Example: Customer Satisfaction Read the example on page 493.Determine the sample size needed to estimate p within 0.03 with 95% confidence. Estimating a Population Proportion The critical value for 95% confidence is z* = 1.96. Since the company president wants a margin of error of no more than 0.03, we need to solve the equation Multiply both sides by square root n and divide both sides by 0.03. Square both sides. Substitute 0.5 for the sample proportion to find the largest ME possible. We round up to 1068 respondents to ensure the margin of error is no more than 0.03 at 95% confidence.

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Read pg. 490 - 494

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