Explain how the graph of the distribution of sample means suggests that the distribution may be approximately normal.

Introduction. Use illustration and history to learn about the Central Limit Theorem. Write about the purpose of the study.

Consider a population that contains values of x equal to 0,1,2,…,18.
Assume that these values occur with equal probability.

**Remark.**
– Search in “Help” : sample.
– Learn about this code for the uniform discrete distribution.

1. Use R to generate 700
samples, each containing 40
measurements from this population (with replacement).
2. Calculate the sample mean x¯ for each of the 700
samples.
3. Construct a relative frequency histogram for the 700
values of x¯
.
4. How does the mean of the sample means compare with the mean of the original distribution ?

5. i) Divide the standard deviation of the original distribution by 40.
ii) How does this result compare with the standard deviation of the sample means distribution ?

6. Explain how the graph of the distribution of sample means suggests that the distribution may be approximately normal.

7. For each sample size n=2,5,10,50,100
, construct a relative frequency histogram of the 700
values of x¯.

8. What changes occur in the histograms as the value of n
increases? What similarities exist?

9. i) Repeat questions 7 and 8 with n=150,200,250,300
ii) Explain how the results above illustrate the Central Limit Theorem.

10. a) Repeat questions 7 and 8 for the 700
values of the sample variance s
2.
b) Repeat questions 7 and 8 for the 700
values of the sample median M.
c) Does it appear that x¯
and M
are unbiased estimators of the population mean?
d) Does it appear that s
is a biased estimator of the population standard deviation σ
?

Reflection.
– Write a paragraph about the purpose of the study, the problem analysed, and your findings.

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