Distributions of Sample Means and Sample Proportions BUSA 2100, Sections 7.0, 7.1, 7.4, 7.5, 7.6

Point Estimates l We don’t expect the sample mean to be exactly equal to the population mean. l Even when two samples are selected from the same population, the two sample means will be different! l In fact, there is an entire distribution of different sample means from the same population. l The distribution of all sample means for samples of a fixed size is called the distribution of sample means.

Distribution of Sample Means l Ex.: A distribution of 25 individual items (population) is: { 40, 50, 55, 59, 62, 64, 65, 66, 67, 68, 69, 70, 70, 70, 71, 72, 73, 74, 75, 76, 78, 81, 85, 90, 100 }. l How many samples of size 4 are possible? l We will take only 12 samples of size 4, and calculate the mean of each sample.

Distribution of Sample Means Example, Page 2.

Distribution of Sample Means (for all samples of a fixed size) l Suppose all samples of size 4 had been chosen.

Central Limit Theorem l Central Limit Theorem: For large values of n, the distribution of sample means becomes normally distributed, regardless of the shape of the distribution of individual items (population). (see text) l When n is larger than 30, that is considered large enough.

Distribution of Sample Means (Summary) l The distribution of sample means for all samples of a fixed size n (from a population with mean mu and standard deviation sigma) has mean mu and standard deviation sigma / sqrt (n). l The symbol for the std. deviation of the sample means is sigma sub X-bar. l Relate to the size of n.

z-Formulas.

Salaries Example l Example: For a population of 2,000 management executives, the salaries are normally distributed with a mean of $56,000 and a standard deviation of $4,200. l A sample of 36 managers is selected and the mean salary is calculated. l What is the probability that the sample mean is within $500 of the population mean? l In other words, what is the probability that the sampling error is <= $500?

Salaries Example, Page 2.

Salaries Example, Page 3.

Distrib. of Sample Proportions l Proportions are always between 0 & 1. l Proportions are binomial. l A sample proportion, p-bar, is a point estimate for the population proportion, p. l For a population (distribution of individual items) with proportion p, the distribution of sample proportions for all samples of a fixed size n has mean = p, and std. dev. = sigma sub p-bar = sqrt [ p * (1 - p) / n ]

Customer Proportion Example l Example: Last year, 30 percent of a company’s mail orders came from first- time customers. l A random sample of 80 mail-order customers is selected and the proportion of first-time customers is calculated.

Proportion Example, Page 2 l What is the probability that the sample proportion is within 4% (.04) of the population proportion?

Proportion Example, Page 3 l Part (b): Same question for n = 250.