Activity 5.1 Introduction to the Distribution of Sample Means
Suppose that you plan to take one random sample with n = 5 from a population with µ = 16 and σ = 5. According to the CLT, what is your best guess for the mean of this one sample?
16
If you click on the "Animated" button again, the program will randomly select another five scores and then put their mean on the Distribution of Sample Means graph. After doing this, how many sample means are on that graph?
2
What is the standard deviation for this population? σ = ________. (Report to the hundredth place.)
2.24
According to the law of large numbers, which of the following options is more likely? (Note: The mean height of women in the United States is 64.5 inches.)
A random sample of two females from the U.S. population will have an average height of70 inches.
Why would we want to make the standard error smaller?
Both of the above.
What shape of a distribution allows you to use the unit normal table (i.e., the z table) to determine the probabilities associated with different z scores?
Normal (bell-shaped)
What shape do you expect for the distribution of sample means?
Normal, bell shaped
The population standard deviation of σ = 5 tells us that the typical distance the individual scores are away from the population mean (µ = 16) is 5. What does the standard deviation of the distribution of sample means (SEM = 2.24) tell us?
The typical distance sample means are from the population mean is 2.24.
Click on the "Clear lower 3" button to clear the graphs. Create a distribution of sample means changing the N to N = 25 and then clicking on the "100,000" button. How close is the standard deviation of the distribution of sample means to the one predicted by the CLT (i.e., the one you just computed)?
Very close (i.e., within .01)
How close are the values to what the central limit theorem predicted in Questions 33 and 34?
Very close (i.e., within .01)
Use the Java Applet to see what happens for skewed distributions of scores. Start by changing the first graph to "Skewed." Then, select 100,000 random samples with a size of 2. Do the same with all of the sample size options (i.e., 5, 10, 16, 20, 25). As the sample size increased, what happened to the shape of the distribution of sample means?
It became more bell shaped.
How is the graph of the distribution of sample means with this larger sample size (n = 25) different from the distribution of sample means when the sample size was smaller (n = 5)?
It is less variable (sample means are closer to the population mean)
How does the standard deviation of the distribution of sample means compare with the standard deviation of the population?
Lower
How did the CLT's predictions for the mean and standard error of the mean compare to those you found by actually building the distribution of sample means and then computing its mean and its standard deviation? Were the predictions made by the CLT perfectly accurate? (Hint: Compare the answers for Questions 20 and 21 to those predicted for Questions 12 and 13, respectively.)
Yes, the answers are identical
Although the graphs look very different, there is one similarity between the graphs. Which of the following is identical for the two graphs?
-Central tendency (i.e., mean)
How could we make the standard error (SEM) smaller? Select all that apply.
-Increase N -Decrease σ
According to the central limit theorem, the standard deviation of the distribution of sample means will be determined by what formula? a. 𝜎/n b. 𝑆𝑆/√SSn−1 c. ∑𝑋/∑Xn
A
How does the mean of the distribution of sample means compare with the mean of the population?
Same
You should recognize that some of these samples represent the population better than others and therefore some have less sampling error than others. Each of the above sample means that are not exactly equal to the population mean of 5 (µ = 5) have sampling error. Which two samples have the most sampling error?
Samples A and P
In Chapter 1, we introduced the idea that sample statistics are rarely exactly equal to the population parameters they are estimating. What is this difference between sample statistics and population parameters called?
Sampling error
According to the central limit theorem, the distribution of sample means
is very often normally distributed.
In general, the __________ the sample, the closer the sample statistic should be to the population parameter.
larger
Sample means are _______ variable than individual scores; therefore, the standard deviation of the distribution of sample means is __________ than the standard deviation of the population.
less, smaller
First, let's look at the shape of the original population, or parent population. There is just one 2, one 4, one 6, and one 8, so the shape of this population's distribution is
not normal
According to the central limit theorem, the mean of the distribution of sample means will be equal to what value?
µ
How does the graph of the distribution of sample means (the frequency distribution you created in Question 15) differ from the graph of the original data (the frequency distribution under Question 14)? (Choose two.)
-Spread (i.e., variability) -Shape
In the previous problems, you were planning to use a sample size of 5. What would happen if you used a larger sample (e.g., 25)? Use the CLT to compute the expected amount of sampling error with a sample size of 25 for the population with a mean of µ = 16 and a standard deviation of σ = 5.
1
Compute the standard deviation of the distribution of sample means. You should be able to use the statistics mode on your calculator to obtain the standard deviation. Because you have the population of all possible sample means, you should use the population equation for the standard deviation. If you choose to do the calculations by hand, be careful about what N you use. When computing the standard deviation of all 16 sample means, N is the number of sample means (i.e., 16).
1.58
Furthermore, according to the central limit theorem, the standard deviation of the distribution of sample means, or the standard error of the mean, will be 𝑆𝐸𝑀𝑃=𝜎/𝑛 So, in this case, the predicted standard error of the mean will equal _________. Now that you've predicted the center and spread of the distribution of sample means for the billionaire example when the n = 2, its time to determine if these predictions are perfectly accurate. (Hint: They will be!) (Report to the hundredth place.) You Answered
1.58
The study of extraversion will use just one of the possible random samples of n = 5 people. We can't know ahead of time which sample mean that will be. However, the central limit theorem allows us to determine what the distribution of all possible random samples with n = 5 would look like. Then, we can determine the most likely sample mean and expected sampling error. If you take all possible random samples of a given size (n = 5) from the population with a mean of 16 and a standard deviation of 5, what do you expect the mean of the distribution of sample means to equal?
16
According to the CLT, how much sampling error do you expect to have (i.e., what value will the standard error of the mean have)?
2.24
Compute the mean of the distribution of sample means (i.e., compute the mean of the 16 sample means in the last column of the table under Question 15). You should be able to use the statistics mode on your calculator to obtain the mean
5
In this case, the predicted mean of the distribution of sample means will equal ________. (The answer is a number.)
5
What is the mean for this population? µ = _______.
5
You should also recognize that all of the above sample means are possible when the researcher randomly selects a sample from the population. Looking at the distribution of sample means, imagine that you randomly pick one sample mean from all the possible sample means. Which sample mean are you most likely to pick? [What is the value of the sample mean that you would most likely pick at random?]
5
Suppose that the average height of women in the United States is 64.5 inches. If you obtained a random sample of 25 women from this population, what is your best guess as to what the mean height will be for that sample?
64.5
To create a distribution of sample means, we need to obtain all possible random samples of a given size from this population. This program doesn't really allow us to take all possible random samples, but it does allow us to take a very large number of samples from the population, and this very large number is a very good approximation of what happens if you take all possible random samples. To take 100,000 samples at a time, click on the "100,000" button. You can even click it 10 times to take 1,000,000 samples. After you do this, what are the mean and the standard deviation of the distribution of sample means? Mean= _______; Standard deviation = _________
Answer 1: 16 Answer 2: 2.24
Next, we are going to use a Java Applet to help you understand sampling distributions with larger (i.e., more realistic) populations and sample sizes. To access the Applet, go to this website: http://onlinestatbook.com/stat_sim/sampling_dist/index.html (Links to an external site.). When you get to this page, you will need to click on the "Begin" button. At the top of the screen, there is a "parent" population of scores. What is the mean and standard deviation for this population distribution of scores (they are located on the left side of the graph)? Mean (µ) = ___________; Standard deviation (σ)=____________.
Answer 1: 16 Answer 2: 5
The Java Applet also allows you to change the shape of the distribution to anything you want. Make a really strange-looking distribution of scores by holding the mouse button and dragging it over the population of scores. The Applet will update the mean and standard deviation of those scores. Record them below: Mean (µ) = ____________; Standard deviation (σ) =______________
Answer 1: 25 Answer 2: 2.93
In all the preceding examples, you started with a normally distributed population of extraversion scores. You are also interested in depression scores. The measure of depression is not normally distributed but skewed with a mean of µ = 8.08 and a standard deviation of σ = 6.22. According to the central limit theorem, what would you expect for the mean and standard deviation of the distribution of sample means for samples of a size of 25 from a positively skewed population with µ = 8.08 and σ = 6.22? Mean= ___________ Standard deviation of the distribution of sample means (SEM) = __________
Answer 1: 8.08 Answer 2: 1.24
Even if a parent population's distribution is skewed, the distribution of the sample means is approximately ____________ and becomes even more so as the sample size _______ .
Answer 1: Normal or bell shaped Answer 2: increases
Take 100,000 random samples of a size of N = 25 from the strange-looking distribution of scores you created. What does the distribution of sample means look like?
Approximately normally distributed (bell shaped)
What do you expect the standard deviation of the distribution of sample means to equal? a. 5 b. 5/√16 c. 5√5
C
To create the distribution of sample means, you need to obtain the means for all possible random samples of sample size N = 2 because that is the sample size used to make our CLT predictions for the mean and standard error of the mean. Because the samples must be random, you must be sure to sample with replacement. Thus, you would choose one score at random, put it back in the population, and then choose again at random. All possible random samples with N = 2 are listed below (labeled A-P). These 16 samples are the population of sample means. Together, they are the distribution of sample means for the population of four billionaires when N = 2. Some of the means are computed for you. Complete the following table by finding the mean for the remaining three samples.
NSample Mean =6 OSample Mean = 7 PSample Mean = 8
According to the central limit theorem, what shape will the distribution of sample means start to approach if all possible random samples with an N = 25 are taken from a positively skewed population with µ = 8.08 and σ = 6.22?
Normally distributed
According to the central limit theorem, what shape do you expect the distribution of sample means to approach when using all possible random samples of a size of 25?
Normally distributed (bell shaped)
Now that you know the µ and the σ, the only other thing you need before you can apply the CLT is to decide on a sample size (N). Again, for illustration purposes, we will use an unusually small sample size of N = 2. According to the central limit theorem, the mean of any distribution of sample means will always be the __________ __________. (two words)
Population Mean
It might help to have a bit of context while you are looking at these distributions, so assume that these numbers represent scores on a test of extraversion, with higher numbers indicating greater extraversion. Suppose that you are going to do a study to determine if extraversion is related to levels of depression. Before starting the study, you want to make sure that you have a good sample with minimal sampling error. What is sampling error?
The difference between a sample mean and the population mean
You can use the Applet to simulate taking one sample at a time from the population. To do this, go to the graph that is labeled "Distribution of means" and select a n = 5. Then, click on the box that says "Animated." This will randomly select five scores from the population and put them in the second graph labeled "Sample Data." The mean of those five scores will then be graphed on the third graph labeled "Distribution of means." After you have taken one sample of five people, what is in the second graph labeled "Sample Data"
The five scores selected randomly from the population
After you have taken one sample of five, what is in the third graph labeled "Distribution of means"?
The sample mean of the five scores
Click on the Animated button several times and make sure that you understand what is happening when you hit the Animated button. After you have several sample means on the Sample Means graph, use the graph to determine which samples have the most sampling error. Remember, sampling error is the discrepancy between a sample statistic and a population parameter. How can you tell which samples have the most sampling error?
The samples furthest from the population mean (µ = 16) have the most sampling error.
Although it is important to understand the idea that larger samples result in less sampling error (i.e., the law of large numbers), you also need to be able to quantify the amount of sampling error for any given study. To do this, you will need to understand the distribution of sample means. Which of the following is the best description of a distribution of sample means?
The set of means for all possible random samples of a given size (n) taken from a population
As the variability in the distribution of sample means decreases, the amount of sampling error ________________.
decrease
