Chapter 18 – Part II

Sampling Distribution for the Sample Mean

Review

One Quantitative Variable– Age– Height

Review

Population parameters– ____________________________________

– ____________________________________

Review

Sample statistics– _______________________________________

– _______________________________________

Example

1994 Senators Population Parameter– _______________________________________

Sample of 10 Senators– _______________________________________

Example

Sample

1

2

3

4

5

SRS characteristics

Value of ___________________________

Changes from sample to sample. Different from population parameter.

– _______________________________

Long Term Behavior of Sample Mean

Repeat taking SRS of size n = 10.– ______________________________________

___________________________________

Sampling Distribution of Sample Mean

Summarize _________________________

__________________________________– Center (Mean)– Spread (Standard Deviation)– Shape

Sampling Distribution of Sample Mean

Mean (Center)

Sampling Distribution for Sample Mean

Standard Deviation (Spread)

Sampling Distribution for Sample Mean

Distribution of Population Values for Quantitative Variable is Normal

Three conditions1. Sample must be random sample

2. Sample must be independent values

3. Sample must be less than 10% of population.

Sampling Distribution for Sample Mean

When Population Values Come from a Normal Distribution

Sampling Distribution for Sample Mean

Distribution of Population Values for Quantitative Variables is non-normal.

Three conditions1. Sample must be random sample

2. Sample must be independent values

3. Sample must be less than 10% of population.

Sampling Distribution for Sample Mean

When Population Values Come from a Non-Normal Distribution.

Central Limit Theorem

As the sample size n increases, the mean of n independent values has a sampling distribution that tends toward a ______________________________.

Central Limit Theorem

Major Result in Statistics!

When Population Values Come from a non- Normal Distribution, Central Limit Theorem still applies when sample size is large.

How large does n need to be?

Depends on shape of population distribution– Symmetric: – Skewed:

68-95-99.7 Rule for Sampling Distribution of Sample Mean

In 68% of all samples, the sample mean will be between

68-95-99.7 Rule for Sampling Distribution of Sample Mean

In 95% of all samples, the sample mean will be between

68-95-99.7 Rule for Sampling Distribution of Sample Mean

In 99.7% of all samples, the sample mean will be between

Example #1

The height of women follows a normal distribution with mean 65.5 inches and standard deviation 2.5 inches.

If you take a sample of size 25 from a large population of women, use the 68-95-99.7 rule to describe the sample mean heights.

Example #1 (cont.)

Example #1 (cont.)

Example #1 (cont.)

Probabilities with the Sample Mean

We can also find probabilities with the sample mean statistic.

Example #1

Ithaca, New York, gets an average of 35.4 inches of rainfall per year with a standard deviation of 4.2 inches. Assume yearly rainfall follows a normal distribution.

Example #1 – cont.

What is the probability a single year will have more than 40 inches of rain?

Example #1 – cont.

Example #1 – cont.

What is the probability that over a four year period the mean rainfall will be less than 30 inches?

Example #1 – cont.

Example #2

Carbon monoxide emissions for a certain kind of car vary with mean 2.9 gm/mi and standard deviation 0.4 gm/mi. A company has 80 cars in its fleet. Estimate the probability that the mean emissions for the fleet is between 2.95 and 3.0 gm/mi.

Example #2 – cont.

Example #2 – cont.

Example #3

Grocery store receipts show that customer purchases are skewed to the right with a mean of $32 and a standard deviation of $20.

Example #3 – cont.

– Can you determine the probability the next customer will spend at least $40?

Example #3 – cont.

– Can you determine the probability the next 50 customers will spend an average of at least $40?

Example #3 – cont.