9.3 Sample Means

Principles Averages are less variable than individual observations. Averages are more normal than individual observations. Example 9.9 Bull or Bear Market? Pg. 591

Parameters and Statistics The mean and standard deviation of a population are parameters. We use Greek letters to write these parameters. The mean and standard deviation calculated from sample data are statistics.

Mean and Standard Deviation of a Sample Mean Suppose that x is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the mean of the sampling distribution of x bar is μx and its standard deviation is

Behavior of x The sample mean x is an unbiased estimator of the population mean μ. The values of x are less spread out for larger samples. You must take a sample four times as large to cut the standard deviation in half. You should only use the recipe for standard deviation when the population is at least 10 times as large as the sample.

Example 9.10 The heights of young women varies approximately according to the N(64.5, 2.5). If we choose one young women at random, the heights we get in repeated choices follow this distribution. That is the distribution of the population is also the distribution of one observation chosen at random. So we can think of the population distribution as a distribution of probabilities, just like a sample distribution. What would be the mean and standard deviation of the sample mean of sample size 10?

Sampling Distribution of a Sample Mean Draw an SRS of size n from a population that has the normal distribution with mean mu and standard deviation sigma. Then the sample mean x has the normal distribution with mean μx and standard deviation

Example 9.11 What is the probability that a randomly selected young women is taller than 66.5 inches? What is the probability that the mean height of a SRS of 10 young women is greater than 66.5 inches?

Homework Exercises p. 595 9.31 – 9.34

Central Limit Theorem Draw an SRS of size n from any population whatsoever with mean μ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x is close to the normal distribution with mean μx and standard deviation Homework p. 601, 9.35 – 9.40

Central Limit Theorem in Action. Right-Skewed Distribution 1 observation Distribution of x for 2 observations Distribution of x for 10 observations Distribution of x for 25 observations

Law of Large Numbers Draw observations at random from any population with finite mean mu. As the number of observations drawn increases, the mean x bar of the observed values gets closer and closer to mu.

Example 9.13 The time that a technician requires to perform preventative maintenance on an AC unit governed by the exponential distribution whose density curve is from the p. The mean time is μ= 1 hour and the standard deviation is σ= 1 hour. Your company has to maintain 70 of these units in an apartment building. You must schedule technicians’ time for a visit to this building. Is it safe to budget an average of 1.1 hours for each unit or should it be 1.25 hours?

Section Exercises Page 603, exercises 9.41 – 9.46