AP STATISTICS LESSON SAMPLE MEANS

ESSENTIAL QUESTION: How are questions involving sample means solved? Objectives:  To find the mean of a sample when μ is unknown.  To find the standard deviation when the σ is unknown.  To solve problems involving sample means.

Sample Means categorical Sample proportions arise most often when we are interested in categorical variables. We then ask questions like “What proportion of U.S. adults have watched Survivor II?” sample means Because sample means are just averages of observations, they are among the most common statistic.

Example 9.9 page 514 Bull Market or Bear Market? Page 515 histograms show:  Averages are less variable than individual observations. More detailed examination of the distribution would point to a second principle:  Averages are more normal than individual observations.

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 is μ x = μ and its standard deviation is σ = σ/ √ n.

 The sample mean x is an unbiased estimate of the population mean μ.  The values of x are less spread out for larger samples.  You should only use the recipe  σ / √ n for the standard deviation of x when the population is at 10 times as large as the sample. Mean and Standard Deviation of a Sample Mean (continued…)

The height of a young women varies approximately according to N(64.5, 2.5 ) distribution. What is the probability that a randomly selected woman is taller than 66.5 inches? What is the probability of an SRS of 10 young women being greater than 66.5? Example 9.10 Page 516 Young Women’s Heights

Sampling Distributions of a Sample Mean from a Normal Population Draw an SRS of size n from a population that has the normal distribution with mean μ and standard deviation σ. Then the sample mean x has the normal distribution N( μ,σ/ √ n ) with mean μ and standard deviation σ/√ n.

The Central Limit Theorem The central limit theorem states that when an infinite number of successive random samples are taken from a population, the distribution of sample means calculated for each sample will become approximately normally distributed with mean μ and standard deviation σ / √ n (N(μ, σ / √ n)) as the sample size (N) becomes larger, irrespective of the shape of the population distribution.

Central Limit Theorem (continued…) 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 N(μ,σ/√ n ) with mean μ and standard deviation σ√ n How large a sample size n is needed for x to be close to normal depends on the population distribution. More observations are required if the shape of the population distribution is far from normal.

Example 9.12 page 521 Exponential Distribution The distribution is strongly right- skewed, and the most probable outcomes are near 0 at one end of the range of possible values. The mean μ of this distribution is 1 and its standard deviation σ is also 1. This particular distribution is called an exponential distribution from the shape of its density curve. As n increases, the shape becomes more normal.

Example 9.13 Page 522 Servicing Air Conditioners The time that a technician requires to perform preventive maintenance on an air conditioning unit is governed by the exponential distribution whose density. The mean time is μ = 1 hour and the standard deviation is σ = 1 hour. Your company operates 70 of these units. What is the probability that their average maintenance time exceeds 50 minutes?

Sampling Distribution The Sampling distribution of a sample mean x has a mean μ and standard deviation σ √ n. The distribution is normal if the population distribution is normal; it is approximately normal for large samples in any case.