The histogram below shows the distribution of ages of pennies at a bank. (a) Des
ID: 3218641 • Letter: T
Question
The histogram below shows the distribution of ages of pennies at a bank.
(a) Describe the distribution.
(b) Sampling distributions for means from simple random samples of 5, 30, and 100 pennies is shown in the histograms below. Describe the shapes of these distributions and comment on whether they look like what you would expect to see based on the Central Limit Theorem.
(c) The mean age of the pennies is 10.44 years, with a standard deviation of 9.2 years. Using the Central Limit Theorem, calculate the means and standard deviations of the distribution of means from random samples of size 5, 30, and 100. Comment on whether the sampling distributions shown in part (b) agree with the values you compute.
5 10 15 20 25 30 10 20 30 40 50 6 8 10 12 14 16 18 7 8 9 10 11 12 13 14 X n 100 n 30Explanation / Answer
4.33 Solution:
(a) The distribution is unimodal and strongly right skewed with a median between 5 and 10 years old. Ages range from 0 to slightly over 50 years old, and the middle 50% of the distribution is roughly between 5 and 15 years old. There are potential outliers on the higher end.
(b) When the sample size is small, the sampling distribution is right skewed, just like the population distribution. As the sample size increases, the sampling distribution gets more unimodal, symmetric, and approaches normality. The variability also decreases. This is consistent with the Central Limit Theorem.
c) for n=5 sample mean=population mean=10.44
sample std deviation=std deviation of population/sqrt(n)' =9.2/sqrt(5)
=4.114
For n=30 sample mean=10.44
sample std dev=9.2/sqrt(30)=1.679
For n=100 sample mean=10.44
sample std dev=std dev/sqrt(n)=9.2/sqrt(100)=9.2/10=0.92
(b) agree with the values you compute as sample size increases for n>30 large samples standard error decrease.
As a sample size is increases, sample variance increases but variance of sample mean decreases and hence precision increases
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