Presentation on theme: "Conditions for Constructing a Confidence Interval"— Presentation transcript:
1 Conditions for Constructing a Confidence Interval Why settle for 95% confidence?Conditions for Constructing a Confidence IntervalRandom: The data come from a well-designed random sample or randomized experiment.Normal: The sampling distribution of the statistic is approximately NormalIndependent: Individual observations are independent and the 10% condition is met.
2 The m&msWhen Mr. Vignolini’s class did the m&ms Activity, they got 107 red m&ms and 144 others. Their point estimate for the unknown proportion p of red m&ms in the population isNow for the challenge: you have about 10 minutes to find a 90% confidence interval for the proportion of red m&ms in the bag!
3 Standard error:We are 90% confident that the interval from to captures the actual proportion of red m&ms in Mr. Vignolini’s container.
4 Mr. Vignolini claims that exactly half of the m&ms in the container are red. Use your result to comment on this claim.The confidence interval in part (a) gives a range of plausible values for the population proportion of red m&ms. Since 0.5 is not contained in the interval, we have reason to doubt Mr. Vignolini’s claim.
5 To find a level C confidence interval, we need to catch the central area C under the standard Normal curve.Our confidence interval has the form:Statistics (critical value).(standard deviation of statistics)
6 Find the critical value (z Find the critical value (z*) and an 80% confidence interval for the proportion of red m&ms in the bag!z* = 1.28
8 One-Sample z Interval for a Population Proportion the critical value for the standard Normal curveUse this interval only when the numbers of successes and failures in the sample are both at least 10 and the population is at least 10 times as large as the sample
9 Binge drinkingAlcohol abuse has been described by college presidents as the number one problem on campus, and it is an important cause of death in young adults. How common is it? A survey of 10,904 randomly selected U.S. college students collected information on drinking behavior and alcohol-related problems. The researchers defined “frequent binge drinking” as having five or more drinks in a row three or more times in the past two weeks. According to this definition, 2486 students were classified as frequent binge drinkers.3. Find the critical value for a 99% confidence interval. Show your method. Then calculate the interval.4. Interpret the interval in context1. Identify the population and the parameter of interestWe are 99% confident that the interval from 21.8% to 23.8% captures the true proportion of U.S. college students who would be classified as binge drinkers2. Check conditions for constructing a confidence interval for the parameter
10 Putting It All Together: The Four-Step Process Confidence Intervals: A Four-Step ProcessState: What parameter do you want to estimate, and at what confidence level?Plan: Identify the appropriate inference method. Check conditionsDo: If the conditions are met, perform calculationsConclude: Interpret your interval in the context of the problem
11 AP EXAM TIP If a free-response question asks you to construct and interpret a confidence interval, you are expected to do the entire four step process. That includes clearly defining the parameter and checking conditions.
12 AP EXAM TIP You may use your calculator to compute a confidence interval on the AP exam. But there’s a risk involved. If you just give the calculator answer with no work, you’ll get either full credit for the “Do” step (if the interval is correct) or no credit (if it’s wrong). We recommend showing the calculation with the appropriate formula and then checking with your calculator. If you opt for the calculator-only method, be sure to name the procedure (e.g., one-proportion z interval) and to give the interval (e.g., to 0.606).
13 Choosing the sample size (n) for Desired Margin of Error when estimating μGiven:Maximum margin of error MEandConfidence interval CSolve the inequality:
14 TattoosSuppose that you wanted to estimate the p = the true proportion of students at your school that have a tattoo with 95% confidence and a margin of error of no more than 0.10.Problem: Determine how many students should be surveyed to estimate p within 0.10 with 95% confidence.z* = 1.96So, we need to survey at least 97 students
15 Customer Satisfaction A company has received complaints about its customer service. The managers 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 satisfied (that is, who choose either “satisfied” or “very satisfied,” the two highest levels on the five-point scale). She decides that she wants the estimate to be within 3% (0.03) at a 95% confidence level. How large a sample is needed?we round up to 1068 respondents to ensure that the margin of error is no more than 3%.
16 Read Section 8.3Exercises on page# 27,31-39 odd, 42, all