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Putting it Together: The Four Step Process

Putting it Together: The Four Step Process. 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. 8.2b h.w: pg 497: 39, 43, 47. Review: Finding a Critical Value

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Putting it Together: The Four Step Process

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  1. Putting it Together:The Four Step Process 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. 8.2b h.w: pg 497: 39, 43, 47

  2. Review: Finding a Critical Value Use Table A to find the critical value z* for an 80% confidence interval. Assume that the Normal condition is met. 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. The closest entry is z = – 1.28. So, the critical value z* for an 80% confidence interval is z* = 1.28. Try invnorm(990)

  3. The Four-Step Process We can use the familiar four-step process whenever a problem asks us to construct and interpret a confidence interval. Estimating a Population Proportion Confidence Intervals: A Four-Step Process State: What parameterdo 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:Interpretyour interval in the context of the problem.

  4. 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.

  5. 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.

  6. 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 pif 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.

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

  8. Ex. Estimating Risky Behavior

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

  10. 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

  11. 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)

  12. = (.4915, 0.5223) • We are 95% confident that the probability of a tossing ahead is between 0.4915 and 0.5223.

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

  14. Choosing the Sample Size In 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. Sample Size for Desired Margin of Error 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:

  15. 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. We round up to 1068 respondents to ensure the margin of error is no more than 0.03 at 95% confidence. Square both sides. Substitute 0.5 for the sample proportion to find the largest ME possible.

  16. Read pg. 490 - 494

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