Using the telephone numbers listed in your local directory as your population, randomly obtain 20 samples of size 3. From each telephone number identified as a source, take the fourth, fifth, and sixth digits.

Calculate the mean of the 20 samples
Draw a histogram showing the 20 sample means. (Use classes -0.5 to 0.5, 0.5 to 1.5, 1.5 to 2.5 and so on).
Describe the distribution of the x-bars that you see in part b (shape of distribution, center, and the amount of dispersion).
Draw 20 more samples and add the 20 new x-bars to the histogram in part b. Describe the distribution that seems to be developing.
Use the empirical rule to test for normality. See the sampling distribution of sample means and the central limit theorem develop from your own data!

To complete this task, follow these steps:

Step 1: Randomly obtain 20 samples of size 3 from the telephone numbers listed in your local directory. For each telephone number identified as a source, take the fourth, fifth, and sixth digits. Make sure the samples are selected without any bias.

Step 2: Calculate the mean of each of the 20 samples. Add up the three digits in each sample and divide the sum by 3 to find the mean. Record these 20 sample means.

Step 3: To draw a histogram showing the 20 sample means, you need to determine the classes for the histogram. Based on the given instruction, the classes will be -0.5 to 0.5, 0.5 to 1.5, 1.5 to 2.5, and so on. Count the number of sample means falling within each class and represent them as bars on the histogram.

Step 4: Describe the distribution of the sample means seen in the histogram. Look at the shape of the distribution, the center (mean), and the amount of dispersion (variability). Is the distribution symmetric, skewed, or neither? Is the center closer to one end or more in the middle? Does it have a narrow or wide spread of values?

Step 5: Draw 20 more samples using the same method as before and calculate the means of these new samples. Add these 20 new sample means to the histogram you created in step 3.

Step 6: Describe the distribution that seems to be developing. Compare it to the initial distribution in part b. Does it appear to have changed? Are there any noticeable patterns or shifts in the distribution?

Step 7: Use the empirical rule to test for normality. The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. Examine the data in your histogram and see if it follows this pattern. If the sample means are normally distributed, the histogram should show a bell-shaped curve as more data is added.

This process allows you to observe the development of the sampling distribution of sample means and the central limit theorem by building your own dataset and analyzing it. Remember that the accuracy of the conclusions drawn from this exercise depends on the randomness and representativeness of the initial telephone number sample.