Sample Means

Sample Means

Calculate the mean and standard deviation of the sampling distribution of a sample mean

Learning Targets

• Calculate the mean and standard deviation of the sampling distribution of a sample mean  $\overline{x}$   and interpret the standard deviation. 

• Explain how the shape of the sampling distribution of  $\overline{x}$   is affected by the shape of the population distribution and the sample size. 

Proportions Vs. Means

We just talked about proportions, but you can also have a sample mean.

Some parts are similar, other parts are different. We are starting with the intro today.

Symbols Vary but Represent Similar Things

• Greek letters refer to populations

• Our Alphabet refers to samples

• If you get them confused - AP will deduct points!

Multiple Choice

These symbols represent the mean and standard deviation for which of the following distributions?

• The Population
• The Sample
• The Sampling Distribution

Multiple Choice

These symbols represent the mean and standard deviation for which of the following distributions?

• The Population Distribution

• The Sample Distribution

• The Sampling Distribution

Multiple Choice

These symbols represent the mean and standard deviation for which of the following distributions?

• The Population
• The Sample
• The Sampling Distribution

What does a Sampling Distribution of Means look like?

The same as any other distribution!

Suppose $\overline{x}$ is the mean of a SRS of size n drawn from a large population with mean $\mu$ and standard deviation  $\sigma$    

Formulas for Mean & Standard Deviation

10% condition will still need to be satisfied in order to use the standard deviation formula.

Example Math

The mean and standard deviation of the population is 120 and 15 respectively. The sample size is 25.

Multiple Choice

The mean and standard deviation of a population are 200 and 20, respectively. Sample size is 25.

What is the mean of the sampling distribution?

• 200

• 16

• 25

• 4

Multiple Choice

The mean and standard deviation of a population are 200 and 20, respectively. Sample size is 25.

What is the standard deviation of the sampling distribution?

• 200

• 16

• 25

• 4

Exploring the Sampling Distribution

• Goto this site: http://onlinestatbook.com/stat_sim/sampling_dist/

• Click Begin

• Take samples (lots!) and figure out if you can ever get a skewed distribution

• Be sure to try varieties of the initial distribution from the pull down menu

Multiple Choice

When you click the animate button under sample, what happens?

• The normal distribution appears on the graph

• The boxes from your sample drift down and form a mean on the next graph

• Each box of the sample is individually plotted on both graphs showing each value

• Reese's Pieces start to be sorted

Multiple Choice

As you continue to add to your sample, what happens to the shape of the distribution of means graph?

• Skewed Left

• Skewed Right

• No Pattern

• Approximately Normal

Open Ended

What is the key to getting a distribution of means that is approximately normal from any initial distribution (could start skewed?)

A few things to note from that applet

• If the population distribution is Normal, then so is the sampling distribution of  $\overline{x}$   
  
  This is true no matter what the sample size is.

• If the population distribution is not Normal, the sampling distribution of  $\overline{x}$   will be approximately Normal when the sample size is sufficiently large $n\ge30$  

Open Ended

How was this version of learning? Compared to Desmos or in general? As in, should I spend time making more of these?

Recap and intro to the Central Limit Theorem

• If the distribution is normal, awesome, proceed as normal

• If the distribution is not normal, then take a large sample - at least 30

• That means if our sample is larger than 30, we can assume the distribution will be normal. One more awesome.

• In a nutshell, that is the Central Limit Theorem

Multiple Choice

What have you observed with the histogram of the sampling distribution of the sample mean?

• The histogram is skewed to the left, regardless of the shape of the population.

• It will tend to have an abnormal distribution, regardless of the shape of the population.

• The bar in the histogram will be equally distributed

• It will tend to have a normal distribution, regardless of the shape of the population.

Multiple Choice

True or False:

Whatever the shape of the distribution of the population, as sample size is increased, the distribution will become approximately normal.

• True

• False

Multiple Choice

What happens to the shape of a sampling distribution of sample means as n increases?

• It becomes narrower and bimodal.
• It becomes narrower and more normal.
• It becomes wider and skewed right. 
• It becomes wider and more normal. 

Wrapping Up

As an intro to the distribution of means, you should made connections to similarities between this and the proportions we did previously. It will be crucial to keep these skills separate. Formulas cannot be interchanged.

Open Ended

Learning Target #1 - Where are you at when it comes to calculating the mean and standard deviation of the sampling distribution of a sample mean? (Slides 12 & 13)

Open Ended

Learning Target #2 - What happens to the shape of the sampling distribution as the sample size changes? Why is the number 30 important? (Slides 18 & 19)