Homework Notes Week 5
I. The Sampling Distribution of a Sample Mean and the Central Limit Theorem
a. Sampling Distribution of a Sample Mean
i. In real situations, statistical studies involve sampling several
individuals then computing numerical summaries of the samples.
ii. Most often, the sample mean, x
, is computed.
iii. If several samples are drawn from a population, they are likely to
have different values for x
. Because the value of x
varies each time a
sample is drawn, x
is a random variable.
iv. For each value of the random variable, x
, we can compute a probability. The
probability distribution of x
is call the sampling distribution of x
.
b. Example:
i. Tetrahedral dice are shaped like a pyramid with four faces. Each
face =2.5corresponds to a number between 1 and 4. Tossing a
tetrahedral die is like sampling a value from the population
{1,2,3,4}. We can easily find the population mean, µ=2.5, and the
population standard deviation σ=1.118.
ii. If a tetrahedral die is tossed three times, the sequence of three
numbers observed is a sample of size 3 drawn with replacement.
The table displays all possible samples of size 3 and their sample
mean x5.
Sampl
e
x5 Sampl
e
x5 Sampl
e
x5 Sampl
e
x5
1,1,1 1.00 2,1,1 1.33 3,1,1 1.67 4,1,1 2.00
1,1,2 1.33 2,1,2 1.67 3,1,2 2.00 4,1,2 2.33
1,1,3 1.67 2,1,3 2.00 3,1,3 2.33 4,1,3 2.67
1,1,4 2.00 2,1,4 2.33 3,1,4 2.67 4,1,4 3.00
1,2,1 1.33 2,2,1 1.67 3,2,1 2.00 4,2,1 2.33
1,2,2 1.67 2,2,2 2.00 3,2,2 2.33 4,2,2 2.67
1,2,3 2.00 2,2,3 2.33 3,2,3 2.67 4,2,3 3.00
1,2,4 2.33 2,2,4 2.67 3,2,4 3.00 4,2,4 3.33
1,3,1 2.67 2,3,1 2.00 3,3,1 2.33 4,3,1 2.67
1,3,2 2.00 2,3,2 2.33 3,3,2 2.67 4,3,2 3.00
1,3,3 2.33 2,3,3 2.67 3,3,3 3.00 4,3,3 3.33
1,3,4 2.67 2,3,4 3.00 3,3,4 3.33 4,3,4 3.67
1,4,1 2.00 2,4,1 2.33 3,4,1 2.67 4,4,1 3.00
1,4,2 2.33 2,4,2 2.67 3,4,2 3.00 4,4,2 3.33
1,4,3 2.67 2,4,3 3.00 3,4,3 3.33 4,4,3 3.67
1,4,4 3.00 2,4,4 3.33 3,4,4 3.67 4,4,4 4.00
iii. The mean of all values of x5 is µx5=2.5 and the standard deviation of
all values of x5 is σx5=0.6455.
iv. Next, we will compare these values to the population mean (2.5)
and population standard deviation (1.118).
v. The mean of the sampling distribution is µx5=2.5, which is the same
as the mean of the population, µ=2.5. This relation always holds.
1. µx5= µ
vi. The standard deviation of the sampling distribution is σx5=0.6455.
This is less than the population standard deviation of σ=1.118. It is
not immediately obvious how these two quantities are related.
1. Note, however, that 0.6455=1.118/√3
2. Recall that sample size is n=3. This suggests that σx5= σ/√n
c. Example:
i. Among students at a certain college, the mean number of hours of
television watched per week is µ=10.5, and the standard deviation
is σ=3.6. A simple random sample of 16 students is chosen for a
study of viewing habits. Let x5 be the mean number of hours of TV
watched by the sampled students. Find the mean µx5 and the
standard deviation σx5 of the sampling distribution.
1. Solution:
a. The mean of the sampling distribution is:
i. µx5=µ=10.5
b. The sample size is n=16. Therefore, the standard
deviation of the sampling distribution is:
i. σx5= σ/√n=3.6/√16=0.9
d. Probability Histogram for a Sampling Distribution
i. In the tetrahedral die example, the population is {1,2,3,4}. When a
die is rolled, each number has the same chance of appearing, 1/4 or
0.25.
ii. The probability histogram for the sampling distribution of x5 with
sample size 3 is obtained from the sampling distribution in the
previous table.
iii. The probability histogram for the sampling distribution looks a lot
like the normal curve, whereas the probability histogram for the
population does not.
iv. Remarkably, it is true that, for any population, if the sample size is
large enough, the sample mean x5 will be approximately normally
distributed. For a symmetric population like the tetrahedral die
population, the sample mean is approximately normally distributed
even for a small sample size like n=3.
e. Sampling Distribution of a Skewed Population
i. If a population is skewed, a larger sample size is necessary for the
sampling distribution of x5 to be approximately normal. Consider the
following probability distribution.
ii. Here are the probability histograms for the sampling distribution of
x5 for samples of size 3, 10, and 30. Note that the shapes of the
distribution begin to approximate a normal curve as the sample size
increases.
iii. The size of the sample needed to obtain approximate normality
depends mostly on the skewness of the population. In practice, a
sample of size n>30 is large enough.
f. The Central Limit Theorem
i. The remarkable fact that the sampling distribution of x5 is
approximately normal for a large sample from any distribution is
part of one of the most used theorems in Statistics, the Central
Limit Theorem.
1. Let x5 be the mean of a large (n>30) simple random sample
from a population with mean µ and standard deviation σ.
Then x5 has an approximately normal distribution, with mean
µx5=µ and standard deviation σx5=σ/√n
ii. The Central Limit Theorem applies for all populations. However, for
symmetric populations, a smaller sample size may suffice.
1. If the population itself is normal, the sample mean x5 will be
normal for any sample size.
II. Determine When it is Appropriate to Use The Central Limit Theorem
a. The Central Limit Theorem
i. Let x5 be the mean of a large (n>30) simple random sample from a
population with mean µ and standard deviation σ. Then x5 has an
approximately normal distribution, with mean µx5=µ and standard
deviation σx5=σ/√n
ii. The Central Limit Theorem applies for all populations. However, for
symmetric populations, a smaller sample size may suffice.
iii. If the population itself is normal, the sample mean x5 will be normal
for any sample size.
iv. Example:
1. A sample of size 45 will be drawn from a population with
mean µ = 15 and standard deviation σ = 3.5. Is it appropriate
to use the normal distribution to find probabilities for x5?
a. Yes, by The Central Limit Theorem, since n > 30, x5 has
approximately normal distribution.
2. A sample of size 8 will be drawn from a normal population
with mean µ= -60 and standard deviation σ= 5. Is it
appropriate to use the normal distribution to find probabilities
for x5?
a. Yes, since the population itself is approximately
normal, x5 has an approximately normal distribution.
3. A sample of size 24 will be drawn from a population with
mean µ=35 and standard deviation σ=1.2. Is it appropriate
to use the normal distribution to find probabilities for x5?
a. No, since the population is not known to be normal and
n is not greater than 30, we cannot be certain that x5
has an approximately normal distribution.
III. Calculating Probabilities Involving a Sample Mean (EXCEL)
a.
b.
c.
d.
i. Answer: 0.1197
e.
f.
g.
h.
i. Answer: 0.00015
i.
IV. Use The Central Limit Theorem to Find a Percentile (Tables and Technology)
a.
b.
c.
d.
e.
V. The Sampling Distribution of a Sample Proportion and The Central Limit Theorem
a.
b.
c.
d.
e.
f.
g.
h.
VI. Calculating Probabilities Involving a Sample Proportion (EXCEL)
a.
b.
c.
d.
VII. Determining Whether the Central Limit Theorem Applies
a.
b.
c.
d.
e.
VIII. Central Limit Theorem for Proportions
a.
b.
c.
d.
IX. Introduction to Confidence Intervals for the Mean (Standard Deviation Known)
a.
b.
c.
d.
e.
f.
g.
X. Finding Critical Values for the Confidence Interval for the Mean (Standard
Deviation Known)
a.
b.
c.
d.
XI. Constructing a Confidence Interval for a Population Mean with Standard
Deviation Known, Application (EXCEL)
a.
b.
c.
d.
e.
XII. Confidence Interval for Population Mean with Standard Deviation Known,
Application
a.
b.
c.
d.
e.
f.
g.
XIII. Introduction to the Student’s t-Distribution
a.
b.
c.
XIV. Finding a Critical Value from the Student’s t-Distribution
a.
b.
c.
d.
e.
XV. Confidence Intervals for a Population Mean with Standard Deviation Unknown
(EXCEL)
a.
b.
c.
d.
e.
f.
g.
h.
XVI. The Relationship Between the Confidence Level and Margin of Error
a.
b.
c.
d.
XVII. Finding the Necessary Sample Size in a Confidence Interval for a Population
Mean
a.
b.
c.
d.
XVIII. Confidence Intervals for a Population Table
a.
b.
c.
d.
e.
f.
XIX. Sample Size Necessary for a Confidence Interval of a Given Width (Proportion)
a.
b.
c.
d.
e.
XX. Confidence Interval for a Proportion (TABLES)
a.
b.
c.
d.
e.
f.
g.
h.
XXI. Sample Size Needed for a Proportion Confidence Interval
a.
b.
c.
d.
e.
f.
g.
h.
