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Sampling Distributions: Understanding the Distribution of the Sample Mean, Study notes of Statistics

This chapter explores the concept of sampling distributions, specifically the distribution of the sample mean. A sampling distribution is a probability distribution of all possible values of the mean computed from a sample of size n from a population with mean µ and standard deviation σ. The properties of the sampling distribution of the mean, including its mean and standard deviation, and how it approaches normality as the sample size increases due to the central limit theorem.

What you will learn

  • What is the distribution of the sample mean called?
  • How does the Central Limit Theorem relate to the sampling distribution of the mean?
  • What is a sampling distribution?

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Uploaded on 09/12/2022

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Sampling Distributions
Chapter 8
8.1 Distribution of the
Sample Mean
A Sampling Distribution of a statistic is a
probability distribution for all possible values of
the statistic computed from a sample of size n.
The sampling distribution of the mean is a
probability distribution of all possible values of
the random variable ‘x bar’ computed from a
sample of size nfrom a population with mean µ
and standard deviation σ.
Example 1: pg. 378- 380
(Pay very close attention!)
Conclusions Based On The
Sampling Distribution of x Bar
1.
The sampling distribution is normally
distributed.
2.
It has mean equal to the mean of the
population.
3.
It has standard deviation less than the
standard deviation of the population.
The Mean and Standard Deviation
of the Sampling Distribution pg.382
Suppose that a simple random sample of size n
is drawn from a large population with mean µ
and standard deviation σ.
The sampling distribution of x bar will have
mean µof x bar = µand standard deviation σof
x bar = σ/ sq. root of n. (see formulas on pg
381)
The standard deviation of the sampling
distribution of x bar, σof x bar, is called the
standard error of the mean.
Example: pg. 389, #12
The Shape of the Sampling
Distribution
If a random variable X is normally
distributed with mean µand standard
deviation σ, then the distribution of the
sample mean, x bar, is normally
distributed with mean µ
xbar
= µand
standard deviation σ
xbar
= σ/n
The Central Limit Theorem Pg. 385
Suppose a random variable X has
population mean µand standard deviation
σand that a random sample size nis
taken from this population. Then the
sampling distribution of x bar becomes
approximately normal as the sample size n
increases. The mean of the distribution is
µ
xbar
= µand the standard deviation is
σ
xbar
= σ/n.
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Sampling Distributions

Chapter 8

8.1 Distribution of the

Sample Mean

A Sampling Distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n. The sampling distribution of the mean is a probability distribution of all possible values of the random variable ‘x bar’ computed from a sample of size n from a population with mean μ and standard deviation σ. Example 1: pg. 378- 380 (Pay very close attention!)

Conclusions Based On The

Sampling Distribution of x Bar

  1. The sampling distribution is normally distributed.
  2. It has mean equal to the mean of the population.
  3. It has standard deviation less than the standard deviation of the population.

The Mean and Standard Deviation

of the Sampling Distribution pg.

Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x bar will have mean μ of x bar = μ and standard deviation σ of x bar = σ / sq. root of n. (see formulas on pg

The standard deviation of the sampling distribution of x bar, σ of x bar, is called the standard error of the mean. Example: pg. 389, #

The Shape of the Sampling

Distribution

If a random variable X is normally distributed with mean μ and standard deviation σ, then the distribution of the sample mean, x bar, is normally distributed with mean μxbar = μ and standard deviation σ (^) xbar = σ/√n

The Central Limit Theorem Pg. 385

Suppose a random variable X has population mean μ and standard deviation σ and that a random sample size n is taken from this population. Then the sampling distribution of x bar becomes approximately normal as the sample size n increases. The mean of the distribution is μxbar = μ and the standard deviation is σ (^) xbar = σ/√n.

2

Central Limit Theorem Characteristics

  1. μxbar = μ
  2. σ (^) xbar = σ/√n
  3. Regardless of the shape of the underlying population, the sampling distribution of x bar becomes approximately normal as the sample size, n, increases. Examples pg. 389-390: 18 and 22