Download Introduction to Chance Models (Section 1.1) Introduction A key step

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b. What does this answer reveal about whether simple random sampling is an unbiased
sampling method? Explain.
Key idea: When we use simple random sampling, the average of the sample means from
different samples equals the population mean and the average of the sample proportions
from different samples equals the population proportion. (This means that these statistics are
unbiased estimators for the corresponding parameter when we use simple random
sampling.)
To really see the long-run pattern in these results, let’s look at many more random samples from
this population. Open the Sampling Words java applet. You will see that the population
distribution contains three variables: the lengths of the words, whether or not a word contains at
least one letter e, and whether or not the word is a noun. For now, select only the length
variable.
17. Check the Show Sampling Options box. Specify 10 in the Sample Size box and press
the Draw Samples button. You will see the 10 selected words appear in the box below.
You will also see 10 darkened dots within the population distribution representing the 10
lengths that you have sampled. The value of sample mean is displayed in the population
box. The sample mean is also plotted in Sample Mean Length plot. Press Draw
Samples again. Did you get the same sample this time? Did you get the same sample
mean?
You should now see two dots in the Sample Mean Length plot, one for each sample
mean. The mean of these two sample means is displayed in the upper right corner of
this plot. Confirm this calculation.
First sample mean:
Second sample mean:
Average:
Change the Number of Samples to 998 (for a total of 1000 samples). Uncheck Animate. Press
the Draw Samples button. (This should generate a dotplot of 1000 values of the sample mean
for 1000 different simple random samples, each consisting of 10 words, selected randomly from
the population of all 268 words in the Gettysburg address.)
18. Describe the resulting distribution of sample means: What are roughly the largest and
smallest values? What is the average of the distribution? What is the standard
deviation? How does the average of the 1000 sample means compare to the population
mean 𝜇 = 4.29?
June 27, 2014
MAA PREP workshop
54