The Practice of StatisticsThird Edition

Chapter 9:Sampling Distributions

Copyright © 2008 by W. H. Freeman & Company

Daniel S. Yates

The distribution of returns for NYSE common (individual) stocks in 1987 Note wide spread; mean return is -3.5

The distribution of returns for portfolios of 5 stocks in 1987Note smaller spread; mean return is still -3.5

Important points from the histograms

-Means of random samples are less variable than individual observations.

-Means of random samples are more Normal than individual observations.

Important points about the behavior of sample means in repeated samples:

-The sample mean xbar is an unbiased estimator of the population mean μ

-Variability of the distribution decreases as n increases

-Must meet rule of thumb one to use standard deviation formula

Note: These statements are true regardless of the sample size (assuming N ≥ 10n)

does not approach μ it equals μ

does not approach it equals n

X

Xn

Height of Young Women For n = 1 ~ N(64.5, 2.5)For n = 10 ~ N(64.5, 0.79)(ignore blue for now)

Height of Young Women For n = 1 ~ N(64.5, 2.5)For n = 10 ~ N(64.5, 0.79)P(X > 66.5) = ?P(xbar > 66.5)= ?

Repeated Measurements

• Common practice to repeat a careful measurement several times and report the average.

• Think of the results of n repeated measurements as an SRS from the population of outcomes we would get if we repeated the measurement forever.

• The average of n results (the sample mean xbar) is less variable than a single measurement.

CLT

• If n is not large, the shape of the sampling distribution of xbar more closely resembles the shape of the original population.

• If population is Normal, shape of sampling distribution will be Normal regardless of sample size.

• If n is large, the shape of sampling distribution will be Normal regardless of population shape.

www.whfreeman.com/tps3e

n = 1 n = 2

n = 3 n = 4

Servicing Air Conditioners

• Time required to perform preventive maintenance has an exponential distn with μ = 1 hour and σ = 1 hour.

• Your company has a contract to maintain 70 units. Should you a lot an average of 1.1 hours to service each unit? Or an average of 1.25 hours?

• CLT says that the time xbar spent working on 70 units is approximately Normally distributed with mean μ = 1 and σ = 0.120.

What is the P(xbar > 1.1)? What is the P(xbar > 1.25)?

Sampling distribution of a sample mean xbar