Section 9.1 Estimating a Population Proportion Objective: Obtain a point estimate for the population proportion; construct and interpret a confidence interval for the population proportion; determine sample size necessary for estimating a population proportion. A point estimate of a population characteristic is a single number that is based on sample data and estimates the value of a parameter. x is used as a point estimate for . s is used as a point estimate for . p̂ is the point estimate for p. pˆ x n A confidence interval for an unknown parameter is an interval of possible values for the parameter. It is constructed so that, with a chosen degree of confidence, the actual value of the parameter will be between the lower and upper bounds of the interval. A level of confidence, c, is a measure of the degree of assurance we have in our point estimate. The confidence level associated with a confidence interval is the success rate of the method used to construct the interval. Theoretically, the value of c may be any number between zero and one. Typical values for c include 0.90, 0.95, and 0.99. z is called the critical value for a confidence level c. (Remember, z is the z-score 2 such that the area under the standard normal curve to the right of z is , so use the positive z-score.) o The determine the critical value for a given level of confidence, 1 c z InvNorm , 0, 1 . For example, if given a 95% level of confidence, 2 2 1 0.95 the critical value will be z InvNorm , 0, 1 1.96 2 2 Confidence interval estimates for the population proportion are of the form point estimate margin of error Confidence Interval for p pˆ E p pˆ E 9.1 - 2 The margin of error, E, determines the width of the interval and depends on : Level of confidence – as the level of confidence increases, the margin of error also increases. Sample size – as the sample size increases, the margin of error decreases (Law of Large Numbers) Standard deviation – an increase in the standard deviation will widen the confidence interval. Constructing a Confidence Interval about p In order to construct a confidence interval for a population proportion, the following condition must be met: ˆ npˆ (1 pˆ ) 10 where p x and the sample size is no more than 5% of the n population size. We will use the TESTS menu on the graphing calculator to construct our confidence intervals. Using TI 83/84 to Construct Confidence Intervals about p Press STAT, highlight TESTS, and select A:1-PropZInt Enter the values of x and n Enter C-Level: Press Enter on Calculate Interpretation of a Confidence Interval for p – “We are _____% confident that the population proportion is between (lower bound) and (upper bound). ” OR “We are _____% confident that the interval actually does contain the true value of p.” Work #1 - 4 9.1 - 3 Determine the Sample Size Necessary for Estimating a population Proportion Compute Minimum Sample Size We know E z 2 pˆ (1 pˆ ) n z Solving for n, we get n pˆ (1 pˆ ) 2 E 2 (round z to three decimal places) Any fractional value of n is always rounded to the next higher whole number. Compute Minimum Sample Size If we have no preliminary estimate for p z n .25 2 E 2 (round z to three decimal places) Any fractional value of n is always rounded to the next higher whole number. Work #5 - 6