# MATH 160 Irvine Valley College Introduction to Statistics Questions

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This extra credit activity will give you an opportunity to better understand the formula for the sample
standard deviation and why it involves division by n-1 instead of n. Use your scientific calculator for
this activity. Please use four decimal places, if necessary.
Consider a family of four people, aged 8, 10, 33, and 37 as a population.
a) Calculate the population variance and standard deviation for the ages. Show your work step by step
for this part.
b) Random samples of size two are drawn with replacement from this population and their ages are recorded.
Find the variance and standard deviation for each sample (using the formula that involves divisions by n-1)
and fill in the 3rd and 4th columns of the table.
Find the variance and standard deviation for each sample by treating the sample as a population (using the
formula that involves divisions by n) and fill in the 5th and 6th columns of the table.
Find the mean for each column and list them in the last row of the table.
Sample
Sample Values
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
SD
(treated as a
sample)
VAR
(treated as a
sample)
SD
(treated as a
population)
VAR
(treated as a
population)
Mean=
Mean=
Mean=
Mean=
8,8
8,10
8,33
8,37
10,8
10,10
10,33
10,37
33,8
33,10
33,33
33,37
37,8
37,10
37,33
37,37
c) The last row of the table shows the Mean of each statistic (over all possible samples of the same size). If the
Mean is equal to the population parameter, the statistic is said to be an unbiased estimator of that parameter.
Which method (dividing by n-1 or dividing by n or neither) resulted in an unbiased estimate of the population
variance? Make sure to justify your answer using the appropriate results from part a) and part b).

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