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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