The shape of the Sampling Distributions

Table of Contents

1. Introduction:
2. Overview of the Applet.
3. When the Population is Normal
4. When the Population is Skewed?

When the Population is Skewed?

When the population was normally distribution we saw that the sample mean also had a normal distribution. This is probably not surprising but what happens when the population is skewed.

Sample size n=5.

Begin by opening the applet:

1. Set the choice of distribution to "Skewed."

1

Mean of the population? Examine the distribution of the population (shown in black). To the left of the "Parent population" you will see a mean (given in blue) and a standard deviation (given in red). What is the MEAN of the parent population here? a) 50 b) 24.79 c) 37.33 d) 18.33 e) 19.42 Correct here the mean of the population is 24.79. Incorrect. Look carefully at the number in blue. Your answer has been saved.

2. Next set the statistic to "Mean" and the sample size to N=5.
3. Click the 1000 button 5 times to produce over 5000 sample means.

2

Shape of the histogram of the means? Examine the histogram of the sample means (the blue histogram). What shape does this histogram have? a) Slightly skewed to the right (right skewed but not as much as the population) b) Strongly skewed to the right (almost the same as the population) c) Symmetric but not at all like a normal distribution. d) Similar to a normal distribution. Correct. The sample means are slightly skewed to the right.Incorrect. Look carefully at the blue histogram. Your answer has been saved.

Sample size n=25

5. Set the choice of distribution to "Skewed."
6. Next set the statistic to "Mean" and the sample size to N=25.
7. Click "Animated" to see the first sample appear on the section marked Sample Data. A blue mark will appear on the panel marked "Distribution of the Means."

3

How many items in the sample? Look at the second graphic. After you hit the "Animated" button some boxes appeared on this second graphic labelled "Sample Data". How many of these boxes appeared? a) 2 b) 4 c) 5 d) 16 e) 25 Correct there are 25 observations in our sample. Incorrect. Is your sample size set to 25? Your answer has been saved.

8. Click the 1000 button 5 times to produce over 5000 sample means.

4

Shape of the histogram of the means? Examine the histogram of the sample means (the blue histogram). What shape does this histogram have? a) Skewed to the right b) Skewed to the left c) Symmetric but not at all like a normal distribution. d) Similar to a normal distribution. Correct. The sample means had a normal distribution.Incorrect. Look carefully at the blue histogram. Your answer has been saved.

Sample size n=100

1. Set the choice of distribution to "Skewed."
2. Next set the statistic to "Mean" and the sample size to N=100.
3. Click "Animated" to see the first sample appear on the section marked Sample Data. A blue mark will appear on the panel marked "Distribution of the Means."
4. Click the 1000 button 5 times to produce over 5000 sample means.

5

Shape of the histogram of the means? Examine the histogram of the sample means (the blue histogram). What shape does this histogram have? a) Skewed to the right b) Skewed to the left c) Symmetric but not at all like a normal distribution. d) Similar to a normal distribution. Correct. The sample means had a normal distribution.Incorrect. Look carefully at the blue histogram. Your answer has been saved.

Summarize the results

6

Which were like a normal distribution? Review the experiments you did with the different sample sizes. In which of the sample sizes did the sample means (blue histogram) resemble a normal distribution? a) n=5 b) n=25 c) n=100 Correct. If the population is skewed the sample mean will have a normal distribution if the sample size is large.Is that all of them? Look back over your experiments and try again.Your answer has been saved.