Distributions of Sample Means Distributions of Sample Means and Sample Proportionsand Sample Proportions

BUSA 2100, Sections 7.0, 7.1, 7.4, 7.5, 7.6

Point EstimatesPoint Estimates We don’t expect the sample mean to be

exactly equal to the population mean. Even when two samples are selected from the

same population, the two sample means will be different!

In fact, there is an entire distribution of different sample means from the same population.

The distribution of all sample means for samples of a fixed size is called the distribution of sample means.

Distribution of Sample Means Distribution of Sample Means 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 } .

How many samples of size 4 are possible?

We will take only 12 samples of size 4, and calculate the mean of each sample.

Distribution of Sample Means Distribution of Sample Means Example, Page 2Example, Page 2

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

chosen.

Central Limit TheoremCentral Limit Theorem 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)

When n is larger than 30, that is considered large enough.

Distribution of Sample Means Distribution of Sample Means (Summary)(Summary)

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).

The symbol for the std. deviation of the sample means is sigma sub X-bar.

Relate to the size of n.

z-Formulasz-Formulas

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Salaries ExampleSalaries Example 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.

A sample of 36 managers is selected and the mean salary is calculated.

What is the probability that the sample mean is within $500 of the population mean?

In other words, what is the probability that the sampling error is <= $500?

Salaries Example, Page 2Salaries Example, Page 2.

Salaries Example, Page 3Salaries Example, Page 3.

Distrib. of Sample ProportionsDistrib. of Sample Proportions Proportions are always between 0 & 1. Proportions are binomial. A sample proportion, p-bar, is a point

estimate for the population proportion, p . 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 ExampleCustomer Proportion Example

Example: Last year, 30 percent of a company’s mail orders came from first-time customers.

A random sample of 80 mail-order customers is selected and the proportion of first-time customers is calculated.

Proportion Example, Page 2Proportion Example, Page 2 What is the probability that the sample

proportion is within 4% (.04) of the population proportion?

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