8 2 Estimating a Population Proportion The Details

8. 2: Estimating a Population Proportion The Details…

The Mystery Proportion �

� 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. The closest entry is z = – 1. 28. z . 07 . 08 . 09 – 1. 3 . 0853 . 0838 . 0823 – 1. 2 . 1020 . 1003 . 0985 – 1. 1 . 1210 . 1190 . 1170 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 90% 95% 99% Tail Area z*

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

� standard error

Checking Conditions �

The Four-Step Process � 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 Xray 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 � margin of error

Choosing the Sample Size �

� Example: Customer Satisfaction Determine the sample size needed to estimate p within 0. 03 with 95% confidence. üThe critical value for 95% confidence is z* = 1. 96. ü 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. 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.

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