Sampling Distributions - Statistical Method - Lecture Slides, Slides of Statistics

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Business Statistics

Introduction to Sampling Distributions

Types of Distribution

• Frequency Distribution
• Probability Distribution
  • Discrete Probability Distribution
    • Binomial Distribution
    • Poisson Distribution
  • Continuous Probability Distribution
    • Normal (Gaussian) Distribution
  • Sampling Distributions

Features of Sampling Distribution

The 4 features of sampling distribution include:

1. The statistic of interest (Proportion, SD, or Mean)
2. Random selection of sample
3. Size of the random sample (very important)
4. The characteristics of the population being sampled.

Sampling Distribution

• A sampling distribution is a

distribution of the possible values of

a statistic for a given size sample

selected from a population

Developing a Sampling Distribution

• Assume there is a population …
• Population size N=
• Random variable, x,

is age of individuals

• Values of x: 18, 20,

22, 24 (years)

7

A (^) B C^ D

Developing a Sampling Distribution

8

. . . 0 18 20 22 24 A B C D Uniform Distribution

P(x)

x

Summary Measures for the Population Distribution:

21 4

18 20 22 24

N

x μ i

= + + + =

= ∑

N

(x μ) σ i 2 =

− = ∑

Sampling Distribution of All Sample Means

10

1st 2nd Observation Obs (^18 20 22 ) 18^18 19 20 20 19 20 21 22 22^20 21 22 24 21 22 23 24 18 19 20 21 22 23 24

0

.

.

.

P(x)

x

Sample Means Distribution

16 Sample Means

_

Developing a Sampling Distribution

(no longer uniform)

Summary Measures of this Sampling Distribution:

11

Developing a Sampling Distribution

N

x

μ (^) x = ∑^ i = + + ++ =

16

(18-21) (19-21) (24- 21)

N

(x μ ) σ

2 2 2

2 i x x

= + + + =

−



Sampling Distribution of x

13

Suppose we have a small finite population consisting of only N=13 numbers: 54, 55, 59, 63, 64, 68, 69, 70, 72, 73, 75, 77, 82 Following Excel charts show (a) the distribution of the population of data, (b) distribution of the sample means for all possible samples of size 2 drawn from the 13 possible numbers

Excel Example

Sampling Error

• Sample Statistics are used to estimate Population Parameters

ex: X is an estimate of the population mean, μ

• Problems:
  • Different samples provide different estimates of the population parameter
  • Sample results have potential variability, thus sampling error exits

Review

• Population mean: Sample Mean:

16

N

x

μ i

where: μ = Population mean x = sample mean xi = Values in the population or sample N = Population size n = sample size

n

x

x = ∑ i

Example

17

If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is

x −μ = 98.6 −99.2 = −0.6 degrees

x

If the Population is Normal

19

If a population is normal with mean μ and

standard deviation σ, the sampling

distribution of is also normally

distributed with and

x

μx = μ n

σ σx =

z-value for Sampling Distribution of x

• Z-value for the sampling distribution of:

20

where: = sample mean = population mean = population standard deviation n = sample size

x μ σ

n

σ

(x μ) z

−

x