 Here’s the formula for a CI for p: p-hat is our unbiased Estimate of p. Z* is called the critical value. I’ll teach you how to calculate that next. This is the standard deviation of p-hat. Notice all the hats – we don’t really ever know p, so we use p-hat to estimate it.

 You will be told the confidence level (i.e. 90%).  Draw a normal curve. Label the confidence level in the MIDDLE.  Notice there is a portion in each tail that is unshaded.  The value of one of those tails is found by subtracting from 1 and then dividing by 2.  Look that value up in the BODY of Table A. Make it positive.  i.e. 90% confidence  = Divide by 2  Look up in Table A. Z* is 1.64 or 1.65.

 Find z* for 94% confidence.  Here are some common z* that you might want to memorize:  90% confidence  z* =  95% confidence  z* = 1.96  99% confidence  z* = 2.575

 Let’s look back at our formula.  The margin of error gets smaller when:  The sample size increases  The confidence level decreases  This is true for all CIs.

 Every inference procedure will have three conditions to check. Today, we are studying constructing a CI for p.  SRS  Normality  Do you remember the normality check for proportions?  Independence  This is called the 10% rule in your book.  Population ≥ 10 n

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.

 One-Sample z Interval for a Population Proportion Calculate and interpret a 90% confidence interval for the proportion of red beads in a container. Your teacher claims 50% of the beads are red. You take a sampleof 251 and 107 are red. Use your interval to comment on this claim. For a 90% confidence level, z* = Conditions? sample proportion = 107/251 = statistic ± (critical value) (standard deviation of the statistic) We are 90% confident that the interval from to captures the actual proportion of red beads in the container. Since this interval gives a range of plausible values for p and since 0.5 is not contained in the interval, we have reason to doubt the claim.

 The margin of error in a confidence interval only accounts for sampling variability.  Other sources of bias can still make our results inaccurate. Examples:  Response bias (did some teens answer the question untruthfully?)  Nonresponse  Wording

 STAT, TESTS, A:”1-PropZInt”  X = Number of successes in the sample  N = Sample Size  C-Level = Confidence Level

 The Gallup Youth Survey asked a SRS of 439 US teens aged 13 to 17 whether they thought young people should wait to have sex until marriage. Of the sample, 246 said “yes.” Construct and interpret a 95% confidence interval for the proportion of all US teens who would say “yes” if asked this question.

 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 oferror. 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

 Example: Customer Satisfaction Read the example on page 493. Determine the sample size needed to estimate p within 0.03 with 95% confidence. The critical value for 95% confidence is z* = 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 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.