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The following example will help us understand The Sampling Distribution of the Mean
Review: The population is the entire collection of all individuals or objects of interest The sample is the portion of the population that is selected for study. Inferential Statistics is the process of using sample information to draw inferences or conclusions about the population.
Consider a population of 5 commuters who are all neighbors. Each commuter was asked how many miles he/she commutes to work each day.
C1 C2 C 3 C4 C
50 miles 84 miles 38 miles 120 miles 48 miles
Find the population mean of the set of data:
Find the population standard deviation :
Look at a sample of two commuters ( n 2 ), and find the mean x to estimate μ. Like: C1 = 50 and C2 = 84, then C1 and C3, then C1 and C4, then C1 and C5… then C4 and C5. How many should we have?
List all possible samples of two commuters and calculate the mean, x , for each sample.
Commuters Data Values (^) Mean, x
The data set of all the sample means in column 3 is called a sampling distribution of the means.
The sampling error is the difference between the value of a sample mean, x , and the population mean, μ.
Sampling error of the mean = = x − μ
For the sample mean 67, calculate the sample error:
For the sample mean 102, calculate the sample error:
Looking back at column 3 of our table, this data set of all the sample means is called a sampling distribution of the means.
Calculate the mean of this data set – we are calculating the mean of the sampling distribution of the means:
The mean of all the sample means of a sampling distribution has special notation: “mu sub x-bar” or X
Calculate the standard deviation of this data set – we are calculating the standard deviation of the sampling distribution of the means:
The standard deviation of the all the sample means of a sampling distribution has special notation:
“sigma sub x-bar” or X
The standard deviation is a measure of how spread out the sample means are from μ.
The mean of the sampling distribution of the means is: X =
What is the population mean which we calculated before? =
So, the mean of the sampling distribution of the means is equal to the mean of the population from which the samples were selected:
X
The standard deviation of the sampling distribution of the means is: X =
What is the population standard deviation which we calculated before? =
The standard deviation of the sampling distribution of the mean, also known as the standard error of the mean, will always be smaller than the population standard deviation and the formula is:
X (^) n
The larger the standard error of the mean is, the more dispersed the samples means are from the population mean. The smaller the standard error, the closer the sample means are to the population mean.
Finite Correction Factor: For a finite population (having a limit), the formula for the standard deviation
of the sampling distribution of the mean, X is: x 1
N n n N
The sampling distribution of the mean is a probability distribution which lists the sample means from all possible samples of the same sample size selected from the same population along with the probability associated with each sample mean.
Notation for the Mean of the Sampling Distribution of the Mean
The mean of the sampling distribution of the mean is denoted by x , read mu sub x bar. Thus, x =
mean of all the sample means of the sampling distribution.
Notation for the Standard Deviation of the Sampling Distribution of the Mean
The standard deviation of the sampling distribution of the mean is denoted by x , read sigma sub x
bar. Thus, x = standard deviation of all the sample means of the sampling distribution.
8.2 – The Mean and Standard Deviation of the Sampling Distribution of the Mean
Mean of the Sampling Distribution of the Mean,^ x
The mean of the sample means of all possible samples of size n is called the mean of the sampling
distribution of the mean, denoted by x. It is equal to the mean of the population from which the
samples were selected. In symbols, this is expressed as:
(^) x
Standard Deviation of the Sampling Distribution of the Mean or Standard Error of the Mean,
denoted by x
The standard error of the mean is the standard deviation of the sample means of all possible samples of
size n of the sampling distribution, denoted by x. The standard error of the mean is equal to the standard
deviation of the population, σ, divided by the square root of the sample size n. That is:
standard error of the mean
population standard deviation sample size
n
x
Interpretation of the Standard Error of the Mean The standard deviation of the sampling distribution of the mean is referred to as the standard error of the mean because it is a measure of how much a sample mean is likely to deviate from the population mean, that is, a measure of the average sampling error. If the standard error of the mean, x , is a small number, then the sampling distribution of the mean has relatively little dispersion and the sample means will be relatively close to the population mean. On the other hand, if the standard error of the mean, x , is a large number, then the sampling distribution of the mean has a relatively large dispersion and the sample means will be relatively far from the population mean.
Example 8.3 pg. 430 – According to a study of TV viewing habits, the average number of hours a
teenager watches MTV per week is 17.9hours with a standard deviation of 3.8hours. If a
sample of 64 teenagers is randomly selected from the population, then determine the mean and standard error of the mean of the sampling distribution of the mean.
Example 8.4 pg. 430-431 – The registrar at a large University states that the mean grade point average of
all the students is 2.95with a population standard deviation of 0.20.
a. Determine the mean and standard error of the sampling distribution if the sampling distribution of the mean consists of all possible sample means from samples of size 25.
b. Determine the mean and standard error of the sampling distribution if the sampling distribution of the mean consists of all possible sample means from samples of size 100.
c. What effect did increasing the sample size have on the mean and standard error of the sampling distribution?
d. In which sampling distribution of the mean (n = 25 or n = 100) would you have a better chance of selecting a sample mean which is closer to the population mean grade point average?
Review Example 8.5 on pg. 432
Sampling from a Non-Normal Population
THEOREM 8.2 – The Central Limit Theorem For any population, the sampling distribution of the mean approaches a normal distribution as the sample size n becomes large. This is true regardless of the shape of the population being randomly sampled.
General Rule for Applying the Central Limit Theorem: The n Greater than 30 Rule For most applications, a sample size n greater than 30 is considered large enough to apply the Central Limit Theorem. Thus, the sampling distribution of the mean can be reasonably approximated by a normal distribution whenever the sample size n is greater than 30.
THEOREM 8.3 Characteristics of the Sampling Distribution of the Mean when Sampling from a Non-Normal Population
If, the following three conditions are satisfied: given any infinite population with mean, μ, and standard deviation, σ, and all possible random samples of size n are selected from the population to form a sampling distribution of the mean, and the sample size, n , is large ( greater than 30 ). then:
- the sampling distribution of the mean is approximately normal.
- the mean of the sampling distribution of the mean is equal to the mean of the population. This is expressed as:
x
- the standard error of the sampling distribution of the mean is equal to the standard deviation of the population divided by the square root of the sample size. This is written as:
n
x
Example 8.8 pg. 442 – GRAM-HAM Bell, a telephone company, states that the average length of time of
long-distance telephone calls is 21.3minutes with a standard deviation of 3.8minutes.
Determine the mean and the standard error of the sampling distribution of the mean and describe the shape of the sampling distribution of the mean when the sample size is: a. n = 36
b. n = 100
c. Compare the sampling distribution of the mean for the sample size of n = 36 and n = 100.
d. If you had to estimate the mean of the population by either randomly selecting a sample of size n = 36 or of n = 100 from the population, then which sample size would give you a better chance of obtaining a smaller sampling error? Explain.
