Download Chapter 6 Sampling Distributions 6.1 Sampling Distributions

Chapter 6
Sampling Distributions
6.1
Sampling Distributions
Definition 6.1 Parameter: A parameter is a numerical descriptive measure of a population. For examples μ and σ are parameters.
Definition 6.2 Statistic: A sample statistic is a numerical descriptive measure of a sample.
For examples x̄ and s are statistics. In real life, most of the cases, μ and σ are unknown.
We estimate μ and σ by x̄ and s respectively.
Definition 6.3 Sampling Distribution: The probability distribution of a statistic that
results when random samples of size n are repeatedly drawn from a given population is
called the sampling distribution of the statistic. The sample mean, median, and standard
deviation and other numerical descriptive measures compute from the sample can be used not
only to describe the sample but also to make inference about the corresponding population
parameters in the form of estimates or tests of hypothesis.
Extra Example 1: Consider a population of N = 5 elements whose values are: 6, 9, 12,
15, and 18.
(a) How many different samples of size n = 3 can be chosen from this population if we
sample without replacement?
(b) List of all possible samples in part (a).
(c) Compute the sample mean (x̄) and sample median (M) for each of the samples and find
the sample distribution of x̄ and median.
(d) Show that the sample mean (x̄) is an unbiased estimator of population mean (μ)). That
is
E(x) = μ
Example 6.1, page 241.
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6.2
Properties of Sampling Distributions: Unbiasedness and Minimum Variance
Definition 6.1: Point Estimator
A point estimator of a population parameter is a rule for formula which tells us how to
use the sample data to calculate a single number that can be used as an estimate of the
population parameter.
x̄ is a point estimator of population mean μ.
Definition 6.2: Unbiased estimator
If the sampling distribution of a sample statistic has mean equal to the population parameter
the statistic is intended to estimate, the statistic is said to be unbiased of the parameter.
Since, E(x̄) = μ, x̄ is an unbiased estimator of population mean μ.
Biased Estimator: If the mean of the sampling distribution is not equal to the parameter,
the statistic is said to be a biased estimate of the parameter.
Example 6.3 on page 246.
Example 6.4 on page 247.
6.3
The Sampling Distribution of a Mean and the Central Limit Theorem
Properties of the Sampling Distribution of x̄
1. Mean of the sampling distribution equals the mean of the population. That is
E(x̄) = μx̄ = μ
2. Standard deviation of the sample mean is
σ
σx̄ = √
n
Note: The standard deviation σx̄ is often referred to as the standard error of the mean.
Theorem 6.1: If a random sample of n observations is selected from a population with a
normal distribution, the sampling distribution of x̄ will be a normal distribution.
Theorem 6.2 (Central limit theorem): If random sample of n observations are drawn
from any population (non-normal, skewed, unknown etc) population with finite mean μ and
standard deviation σ, then when n is large (n ≥ 30), the sampling distribution of the sample
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mean x̄ is approximately normally distrubuted, with mean and standard deviation (SD) are
respectively
σ
E(x̄) = μx̄ = μ and σx̄ = √ .
n
That is, for large n,
σ
x̄ ≈ N μ, √
n
The approximation will become more and more accurate as n becomes larger and larger.
The symbol ≈ ” stands for approximately distributed.
Comments: When the population distribution of x (measurements) is symmetrical about
the mean μ, the CLT will apply very well to small sample size, n = 10. However, if the
population is skewed, the larger the sample sizes are required to yield an effective approximation to the distribution of x̄ by the normal probability distribution. As a rule of thumb,
large sample size implies, n ≥ 30.
See Figure 6.10, page 252.
Extra Example 2: Assume that the first year students of FIU has a mean GPA, μ = 2.50
and standard deviation, σ = 1.0. Suppose you have selected a random sample of size n = 50
students from these students and the GPA of each student is determined. Let x̄ be the mean
GPA level of the students.
(a) What would be approximate distribution of x̄ according to the central limit theorem?
(b) What is the approximate probability that x̄ is less than 2.25?
(c) What is the approximate probability that x̄ is between 2.25 and 2.75?
Example 6.7, page 252.
Example 6.8, page 254.
Exercise 6.28, page 257.
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