How accurate is a sample mean?

WISE Exercise 1

How accurate is a sample mean?

WISE Exercise 1

It will be helpful to you if you print this web page before doing the assignment.

A friend of yours developed a scale to measure Life Satisfaction. For the population of American adults the scale has a normal distribution with a mean of .50 and a standard deviation of .20. She plans to measure Life Satisfaction for various groups of people, and she is very interested in knowing how accurate her sample means are likely to be. She is coming to you for advice. This exercise is designed to help you prepare for her questions. (You may want to print a copy of this section.)

Review - normal distributions: Life Satisfaction (n=1)

We begin with a single score drawn from this normal distribution. (You don’t need the applet yet.) A brief review of the normal distribution is attached. You may go there first if you wish.

Q1. What is the probability that a randomly selected American adult has a Life Satisfaction score within .05 of the population mean? (i.e., in the range .45 to .55)

Use the figure to make an approximation. A good way to make an approximation here is to guess what percent of the total area under the curve falls between .45 and .55. Remember, the total area is 1.0 (or 100%).

Probability of a score from .45 to .55 is about ______.

Now, before going on, solve for the exact answer. (The answer is about 20%.) You will need to use a table of probabilities for the standardized normal distribution. The exact answer and detailed calculations are in the answer section. Show your calculations in the space below.

Sampling distribution of the mean: Life Satisfaction (n=100)

For this exercise we will use the Sampling Distribution Applet. If you wish, you may consult the attached description of the various features in the applet now. If you haven’t accessed the applet at this point, double click Netscape.

Our researcher friend plans to draw a sample of 100 people. To simulate this, select the following options: normal, n=100, show sample data (no other ‘show’ options for now). If you have questions about any of the options, please consult the attached description of the applet for clarification.

Click on Draw a sample, and write the value displayed for Last mean = in the first space below. Then click on Draw a sample nine more times and record each mean below.

____    ____    ____    ____     ____    ____    ____     ____    ____    ____

How many of your 10 sample means fell outside of the range .45 to .55? _______

Click on Show obtained means to see a display of all ten means.

Notice how tightly clustered the obtained means are compared to the individual scores.

Click Draw 100 samples to see the means for 100 different samples, each of size n=100.

Click Show sampling distribution of the mean to see how closely the observed sample means match the actual distribution of all possible means for samples of size n=100. If you wish, you may consult the attached review of the Central Limit Theorem now.

Q2. What is the probability that a randomly selected sample of n=100 American adults has a mean Life Satisfaction score within .05 of the population mean?

First, make an estimate of the answer. _______

You can also estimate the answer by counting the number of sample means out of 100 that fall within the range .45 to .55. It may be easier to see if you turn off Show sampling distribution of the mean. You don’t need to get an exact count, but we see that when we draw a sample with n=100, it is unlikely that the sample mean is in error by more than .05 as an estimate of the mean Life Satisfaction for the population (i.e., it is unlikely to find a sample mean that deviates far from the mean). Were any of your 100 sample means .05 or more away from the population mean?

Now solve for the exact answer, using z-scores, before going on. The answer is over 98%. The exact answer and detailed calculations are in the answer section. Show your calculations below.

Hint: Consult the attached review of the Central Limit Theorem for more information, including the formula for the standard error of the mean (standard deviation of the sampling distribution).

## Sampling distribution of the mean with samples of n=25

Our researcher wishes to know how accurate the sample mean is likely to be if she samples 25 people. To simulate this, select the following options: normal, n=25, show sample data (no other ‘show’ options).

Click on Draw a sample, and write the value shown for Last mean in the first space below. Then click on Draw a sample nine more times and record each mean below.

____ ____ ____ ____ ____ ____ ____ ____ ____ ____

How many of your 10 sample means fell outside of the range .45 to .55? _______

Click on Show obtained means to see all ten means displayed. Notice how tightly clustered the obtained means are compared to the individual scores.

Click Draw 100 samples to see the means for 100 different samples, each of size n=25.

Click Show sampling distribution of the mean to see how closely the distribution of 100 observed sample means matches the actual distribution of possible means of size n=25.

Q3. What is the probability that a randomly selected sample of n=25 American adults has a mean Life Satisfaction score within .05 of the population mean?

First, estimate the answer by examining your ten sample means, the displays of 100 sample means with n=25 for each mean, and the sampling distribution of the mean. What is your best estimate based on these observations? _______

The exact answer is a little less than 80%. What is the exact answer? ________ A detailed solution is in the answer section, but try it on your own before consulting the solution. Show your calculations below.

Sampling distribution of the mean with n=5

Your researcher friend has considered using sample sizes of only five people. She will ask you to explain the advantages and disadvantages of this plan.

To begin, select the following options: normal, n=5, show sample data (no other ‘show’ options).

Click on Draw a sample, and record the value shown for Last mean in the first space below. Then click on Draw a sample nine more times and record each mean below.

____    ____    ____    ____     ____    ____    ____     ____    ____    ____

How many of your 10 sample means fell outside of the range .45 to .55? _______

Click on Show obtained means to see all ten means displayed.

Notice how tightly clustered the obtained means are compared to the individual scores, and to the distributions of means when n=100.

Click Draw 100 samples to see the means for 100 different samples, each of size n=5.

Click Show sampling distribution of the mean to see how closely the observed sample means match the actual distribution of possible means of size n=5.

How does this sampling distribution of the mean (for n=5) compare to the sampling distribution of the mean for n=100? Indicate your answer below.

Q3. What is the probability that a randomly selected sample of n=5 American adults has a mean Life Satisfaction score within .05 of the population mean?

You can estimate the answer by examining your ten sample means and the displays of 100 sample means with n=5 for each mean. What is your estimate based on these observations? _______

The answer is a little more than 40%. What is the exact answer? ________

Your researcher friend says "we know that for any population, the best estimate of the mean is the sample mean -- therefore, it shouldn't matter what size sample I use, right? Since that is the case, I'll use a sample of n=5 as this will save a good deal of time and money." What do you tell your friend? In your answer, include information from the exercises you have just completed.