Estimating Means and Proportions Using Sample Means and Proportions To Make Inferences About Population Parameters Estimating Means • Almost Always We Do Not Know the Value of the Population Parameters of Interest. Hence, We Must Estimate Their Values Based on Sample Information. • Estimators: – X-bar – P -hat – s2 Sampling Distribution Of The Mean • Shape: Approximately Normal with sufficiently large sample size (n). • Mean: The mean of this distribution is the same as the mean of the population from which the sample is drawn. • Variance: The variance of this distribution is equal to the population variance divided by the sample size (n). • Justification: Central Limit Theorem. • Implications: If we draw a random sample of sufficient size we can estimate the mean and variance and make probability statements about X-bar. Vending Machine Example Revisited • Recall the cup was 7oz., the population mean and standard deviation was 5oz. and .75oz. respectively. • Suppose 30 students decide to get coffee during the break. What is the probability that the average cup will be less than 4 oz. ? Vending Machine Continued • Step 1: P (X-bar < 4oz.) • Step 2: Z = (4 - 5)/.75/(sq.. root of 30) – Z = -7.3 • Draw the picture • P (Z < -7.3) < .001 • Interpretation: It’s very unlikely that that the average of 30 Introduction To Point And Interval Estimation • Suppose we do not know the average amount of coffee dispensed but we do know the standard deviation (.75oz.). • We draw a sample of 30 students and compute the sample mean. It turns out to be 5.25oz. • Point Estimate: Our best estimate of the population mean is the sample mean 5.25oz. Computing An Interval Estimate • Suppose we want to express that we’re not really sure what the population mean is and would rather put an upper and lower bound on our estimate. • Confidence Interval (see formula): Inferences About Proportions • Suppose that X = the number of occurrences of a particular event of interest (e.g., people voting for a candidate, coin turning up heads, people buying a product). • P-hat: p = x/n • Mean of X = np • Variance of X = npq (q = 1-p) • Mean of p-hat = P • Variance of p-hat = pq/n Inferences About Proportions Continued • For large sample size (n) p-hat is approximately normally distributed (Central Limit Theorem). • Z = (p-hat - P)/sq. root of PQ/n • Interval Estimate: See formula given in class What Proportion Of Prospects Will Buy • Suppose we are selling insurance and we approach 40 prospects on a given day. Sales records indicate that on average about 20% of prospects buy ( that is P = .2). • What is the chance that the proportion of these 40 prospects that buy will be less than 10%? • Suppose we were actually able to sell 25%. What is the 95% Confidence Interval in this situation?