AP Statistics: Chapter 8, lesson 2: Estimating a population proportion
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1 Activity 1: Which way will the Hershey s kiss land? When you toss a Hershey Kiss, it sometimes lands flat and sometimes lands on its side. What proportion of tosses will land flat? Each group of four selects a random sample of 50 Hershey s Kisses to bring back to their desks. Toss the 50 Kisses and then calculate the proportion that land flat. Let = the proportion of the Kisses that land flat. 1. What is the value of your point estimate for the true proportion that land flat? 2. Identify the population, parameter, sample and statistic. Population: Parameter: Sample: Statistic: 3. Was the sample a random sample? Why is this important? 4. What is the formula for calculating the standard deviation of the sampling distribution of? 5. What condition must be met to use this formula? Has it been met? 6. We don t know the value of p (that s the whole point of a confidence interval) so we will use instead. Calculate the standard deviation. 7. Would it be appropriate to use a normal distribution to model the sampling distribution of answer.? Justify your
2 8. In a normal distribution, 95% of the data lies within approximately standard deviations of the mean, but this is not the exact number of standard deviations. The critical value is a multiplier that makes the interval wide enough to have the stated capture rate; it depends on both the confidence level C and the sampling distribution of the statistic. Use table A to find these critical values: 80% of the data lies within standard deviations of the mean 90% of the data lies within standard deviations of the mean 95% of the data lies within standard deviations of the mean 99% of the data lies within standard deviations of the mean 9. Calculate the margin of error for a 95% interval by multiplying the critical value and standard deviation you found. Show your work. 10. Find the 95% confidence interval using point estimate +/- margin of error. 11. Add your interval to the graph on the board. Sketch the graph below. 12. What do you think is the true proportion of kisses that land flat is?
3 Example 1: Finding critical values for z Use Table A to find the critical value z* for a 96% confidence interval. Assume that the Large Counts condition is met. Example 2: Sleep Awareness Week begins in the spring with the release of the National Sleep Foundation s annual poll of U.S. sleep habits and ends with the beginning of daylight savings time, when most people lose an hour of sleep. In the foundation s random sample of 1029 U.S. adults, 48% reported that they often or always got enough sleep during the last 7 nights. a. Identify the parameter of interest. b. Check if the conditions for constructing a confidence interval for p are met. c. Find the critical value for a 98% confidence interval. Then calculate the interval. d. Interpret the interval in context.
4 Activity 2: How much is water? What proportion of the Earth is covered by water? We will investigate this question by taking a random sample of locations on the globe. 1. How many locations did your class sample? How many locations were water? 2. Calculate the proportion of locations from your sample that are water. ˆp = 3. Construct a 95% confidence interval to estimate the proportion of the Earth that is water. STATE: State the parameter you want to estimate and the confidence level. Parameter: Confidence level: PLAN: Identify the appropriate inference method and check conditions. Name of procedure: Check conditions: DO: If the conditions are met, perform the calculations. General Formula for any confidence interval: Specific Formula for this confidence interval: Plug numbers into the formula: Answer: CONCLUDE: Interpret your interval in the context of the problem. Interpret:
5 Example 3: Proper procedure for confidence intervals Pennies Mrs. Quinn s class took an SRS of 102 pennies and discovered that 57 of the pennies were more than 10 years old. a. Calculate and interpret a 99% confidence interval for p = the true proportion of pennies from the collection that are more than 10 years old. b. Is it plausible that exactly 60% of all the pennies in the collection are more than 10 years old? Explain. Example 4: Sample size Customer Service 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 value of interest is the proportion p of customers who are satisfied with the company s customer service. She decides that she wants the estimate to be within 3 percentage points (0.03) at a 95% confidence level. 1. Using a conservative estimate for p, how large of a sample is needed? 2. In the company s prior-year survey, 80% of customers surveyed said they were satisfied. Using this value as a guess for p, find the sample size needed for a margin of error of at most 3 percentage points with 95% confidence. How does this compare with the required sample size from question #1? 3. What if the company president demands 99% confidence instead of 95% confidence? Would this require a smaller or larger sample size, assuming everything else remains the same? Explain your answer.
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