Assume that 5 students have the following ages: 17, 19, 22, 26, and 20. Assume that random samples of size n=2 are selected with replacement (the same student could be chosen twice). a. Make a list of all possible samples. b. Find the mean of each sample and construct a table and histogram of the sampling distribution of the means. c. Compare the mean of the population to the mean of the sampling distribution of the mean. d. Do the sample means target the value of the population mean?
Assume that 5 students have the following ages: 17, 19, 22, 26, and 20. Assume that random samples of size n=2 are selected with replacement (the same student could be chosen twice). a. Make a list of all possible samples. b. Find the mean of each sample and construct a table and histogram of the sampling distribution of the means. c. Compare the mean of the population to the mean of the sampling distribution of the mean. d. Do the sample means target the value of the population mean?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:In Figure 2, the die was rolled twice and the process was repeated 500 times. The histogram depicts the sampling
distribution of the means which is looking more bell-shaped. This gets better in Figure 3 and Figure 4 where the number
of rolls (sample size) increased to 10 and 30, respectively. The histograms for each set of averages show that as the
sample size, or number of rolls, increases, the sampling distributions of the means comes closer to resembling a normal
distribution. In addition, the variation of the sample means decreases as the sample size increases.
The Central Limit Theorem states that for a large enough n, the population mean u can be approximated by a normal
distribution with mean i and standard deviation The population mean for a six-sided die is (1+2+3+4+5+6)/6=3.5 and the
population standard deviation is 1.708. Thus, if the theorem holds true, the mean of the thirty averages should
= 0.3 Using the die we rolled via simulation on Microsoft Excel, the mean of
the 30 averages is 3.49 and the standard deviation is 0.30 which are very close to the calculated approximations.
1.708
be about 3.5 with standard deviation -
V30
АCTIVITY
Assume that 5 students have the following ages: 17, 19, 22, 26, and 20. Assume that random samples
of size n=2 are selected with replacement (the same student could be chosen twice).
a. Make a list of all possible samples.
b. Find the mean of each sample and construct a table and histogram of the sampling distribution of the
means.
c. Compare the mean of the population to the mean of the sampling distribution of the mean.
d. Do the sample means target the value of the population mean?
Transcribed Image Text:Illustrating the Central Unit Theorem
Central Limit Theorem: If random samples of size n are drawn from a population, then as n becomes larger, the
sampling distribution of the mean approaches the normal distribution, regardless of the shape of the population
distribution.
Illustrative Example
The Central Limit Theorem can be illustrated by carrying out an experiment using a die. To get
accurate representation of the population distribution, let’s roll a die 500 times. Figure 1 shows the resultof
this scenario, made possible by simulation using Microsoft Excel. As expected, we can see that the
distribution looks fairly flat, and not a normal distribution.
Frequency Distribution of 1 Roll of
Sampling Distribution of the Means
(2 Rolls)
Die
I 15 25 as 4 4sS SS
Value on Die
Sample Mean
Figure 1. Histogram reflecting the frequency of rolling a die 500
Figure 2. Histogram of the sampling distribution of the means
times.
for rolling the die twice.
Sampling Distribution of the Means
(10 Rolls)
Sampling Distribution of the Means
(30 Rolls)
0.14
0.12
0.06
R.04
151.7 192.121 2
29113335271941 4345 4749 5.1 5.3555759
Sample Mean
Sample Mean
Figure 3. Histogram of the sampling distribution of the
Figure 4. Histogram of the sampling distribution of the
means for rolling the die 10 times.
means for rolling the die 30 times.
Probability
Frequency
Probability
Probability
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