8.2: Estimating a Population Proportion The Details…

The Mystery Proportion I have chosen a population proportion, 𝑝, and stored it in my calculator. I will generate a value of 𝑝 for a random sample of size 100. Check to see if the conditions (regarding shape, center, spread) are met. Work with your neighbor to find and interpret a 95% confidence interval. Now for the challenge Work with your neighbor to find a 90% confidence interval.

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. 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. z .07 .08 .09 – 1.3 .0853 .0838 .0823 – 1.2 .1020 .1003 .0985 – 1.1 .1210 .1190 .1170 The closest entry is z = – 1.28. So, the critical value z* for an 80% confidence interval is z* = 1.28.

Common Critical Values, z* z* with probability p lying to its right under the standard normal curve is called the upper p critical value. Some common values for z*: Confidence Level, C Tail Area z* 90% 95% 99%

Critical Values, z* Use Table A and a sketch of the Normal curve to find the critical value, z*, for… an 80% confidence interval a 72% confidence interval

Constructing a Confidence Interval for 𝑝 We can use the general formula we learned in 8.1: statistic ± (critical value)∙(standard deviation of statistic) Plug in what we know for proportions: 𝑝 ±𝑧*∙ 𝑝 (1− 𝑝 ) 𝑛 standard error

Checking Conditions Shape: Normal? 𝑛𝑝≥10 AND 𝑛 1−𝑝 ≥10 Center: Random – so we have an unbiased estimator? 𝜇 𝑝 =𝑝 Spread: 10% Condition? 𝜎 𝑝 = 𝑝(1−𝑝) 𝑛 Note: We do NOT know 𝑝 (that’s the whole point), so we will use 𝑝 instead. Calculating the standard deviation using 𝑝 instead of 𝑝 is called finding the standard error of the statistic.

EVERY TIME????? … YUP, for the rest of the year! The Four-Step Process State: What parameter do you want to estimate, and at what confidence level? (Basically, define 𝑝.) 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. EVERY TIME????? … YUP, for the rest of the year!

Use the Four-Step Process Consider a researcher wishing to estimate the proportion of X-ray machines that malfunction and produce excess radiation. A random sample of 40 machines is taken and 12 of the machines malfunction. Estimate the true proportion of X-ray machines that malfunction. Use a 95% confidence interval.

Use the Four-Step Process In what proportion of British couples is the wife taller than the husband? In a sample of 200 British couples, the wife was taller than the husband in only 10 couples. Estimate the true proportion of British couples in which the wife is taller than the husband. Use an 80% confidence interval.

Choosing the Sample Size Recall, in a confidence interval for proportions: 𝑝 ±𝑧* 𝑝 (1− 𝑝 ) 𝑛 What if we want a specific 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: 𝑧* 𝑝 (1− 𝑝 ) 𝑛 ≤𝑀𝐸 margin of error

Choosing the Sample Size 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 service, measured on a five-point scale. The president wants to estimate the proportion 𝑝 of customers who are satisfied. She decides that she wants the estimate to be within 3% at a 95% confidence level. How large a sample is needed?

Multiply both sides by square root n and divide both sides by 0.03. Example: Customer Satisfaction Determine the sample size needed to estimate p within 0.03 with 95% confidence. 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.

Homework Pg. 496: #35, 37, 41, 43, 47 Take-home FRAPPY (2010 #3)