STAT 104 - Chapter 15 - Sampling Distribution

1. The population consists of all individuals of interest.

2. A sample consists of the individuals that are actually observed.

1. A parameter is a number that describes a population.

ex. μ and σ represent the mean and standard deviation of a population. These numbers are parameters

Heights of young women have a mean of 64.2 ad std dev of 2.8 inches. 64.2 and 2.8 are parameters because they describe the population of all young women. 

A statistic is a number that describes a sample.

ex. x̅ and s represent the mean and standard deviation of a sample. These numbers are statistics.

2. Minitab always thinks we are working with samples, even if its a population. So they will call it a statistic even if its a parameter.

So BE CAREFUL!

1. The numbers for x̅ and s will change.

2. Since numbers calculated from samples vary from one sample to another, they are variables. 

3. No, we cannot (probably because there would be too many). 

4. We can use theory to help us visualize the histogram of all possible values of x̅. The shape of this histogram is called the distribution of the sample mean.

1. Because they have different samples!

2. BIG IDEA - The average of samples varies depending on which sample we have.

σx̅ - Sigma of x bar is the standard deviation of the mean of x̅.

In other words, it is the standard deviation of all possible samples in a population.

μx̅ - Mean of x bar is the mean of all possible values of x̅.

In other words, it is the mean of all possible samples in a population.

RULE 1: μx̅ = μ

RULE 2: σx̅ = σ / √n

*n = the number of individuals in each of the possible samples. So in other words, the sample size. 

RULE 3: The shape of the distribution of x̅ is exactly normal if the population from which the samples are drawn is normal. 

RULE 4: The shape of the distribution of x̅ is approximately normal if the sample size (n) is large enough (n ≥ 30).

*If there is no information, then the shape is 'unknown.'

1. Rule 4 is called the Central Limit Theorem,

It is important because it allows us to work with skewed distributions as well as populations whose distributions are unknown.

** 8 STEPS **

STEPS 1,2 and 3: Find the mean, standard distribution and shape of the population.

ex. The population has a mean of 64.2, a standard distribution of 2.8 and a normal shape.

STEPS 4, 5 and 6: Find the mean, standard deviation and shape of the distribution of x̅

ex. The distribution of x̅ (from n = 20 heights) has a mean (μx̅) of 64.2, a standard deviation (σx̅) of 2.8 ÷ √20 = 0.6261 and a shape that is exactly normal. 

STEP 7: Draw distribution of x̅

• ex

STEP 8: Calculate the percentage of samples of young women's heights 66 inches and over by either standardizing and using z-tables, or by using Minitab. 

• ex. 
• z = x - μ ÷ σ

z = 66 - 64.2 ÷ 0.6261 (σx̅) 

z = 2.8002

z converted to % of distribution x̅ = 0.9974

0.9974 x 100% 

99.74%

* Because x tables work only on the left, and you want to find the distribution of x̅ OVER 66 inches, you need to subtract from 100 or use the opposite negative z number.

100 - 99.74% = 

0.26%

• EXAMPLE USING MINITAB

0.9979 x 100 = 99.79%

*Because Minitab only works on the left, you need to subtract 99.79 from 100

100 - 99.79% = 

0.21%

As the sample size increases, the sample mean (x̅) is more likely to be close to μ

Consequently, in large samples, x̅ is likely to be close to μ