1. Business Statistics Topic 5
Normal probability distribution &
sampling distribution

2. Business Statistics:Topic 5
Learning Objectives
By the end of this topic you will be able to:
Describe the characteristics of the normal distribution
Solve problems involving the normal distribution
Solve problems involving sampling distributions
Business Statistics:Topic 5

3. Why Normal Distribution?
It describes many random processes or continuous phenomena
A wide variety of naturally occurring random variables such as heights and weights of all creatures are distributed evenly around a central value, average, or norm (hence, the name normal distribution). Although the distributions are only approximately normal, they are usually quite close.
It can be used to approximate discrete probability distributions such as binomial and Poisson
It is the basis for classical statistical inference
Business Statistics:Topic 5

4. Properties of Normal distribution
Continuous random variable
‘Bell-shaped’ & symmetrical
Mean, median, mode are equal
Area under the curve is 1
Business Statistics:Topic 5

5. Business Statistics:Topic 5
Standard Normal Curve
Business Statistics:Topic 5

6. Business Statistics:Topic 5
正态分布（normal distribution）是数理统计中的一种重要的理论分布，是许多统计方法的理论基础。正态分布有两个参数，μ和σ，决定了正态分布的位置和形态。
为了应用方便，常将一般的正态变量X通过Z变换[(X-μ)/σ]转化成标准正态变量u，以使原来各种形态的正态分布都转换为μ=0，σ=1的标准正态分布（standard normal distribution）,亦称Z分布。
Business Statistics:Topic 5

7. Continuous Probability
The probability of a continuous variable being between ‘b’ & ‘c’is the area under curve between them.
Business Statistics:Topic 5

8. Area Under Standard Normal Curve
Area between Z=0 and Z =1.02
0.4357
0.4345
0.4332
1.5
0.3461
0.3438
0.3413
1.0
0.3212
0.3186
0.3159
0.9
.02
.01
.00 Z
1.02
Business Statistics:Topic 5

9. Effect of Varying the Parameters (m & s)
Business Statistics:Topic 5

10. Effect of Varying the Parameters
Normal distributions differ by mean & standard deviation
Each distribution would require its own table.
That’s an infinite number!
Business Statistics:Topic 5

11. Standardizing Normal Distribution
Standardized Normal Distribution
Normal Distribution
This way you only need to use one table
Business Statistics:Topic 5

12. Business Statistics:Topic 5
Example
Standardized Normal Distribution
Normal Distribution
Same Area
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13. Standard Normal Probability Table
0.4357
0.4345
0.4332
1.5
0.3461
0.3438
0.3413
1.0
0.3212
0.3186
0.3159
0.9
.02
.01
.00 Z
Business Statistics:Topic 5

14. Business Statistics:Topic 5
Examples
A venture capital company feels that the rate of return (X) on a proposed investment is approximately normally distributed, with mean of 30% and standard deviation of 10%.
Business Statistics:Topic 5

15. Business Statistics:Topic 5
Example
The probability that the return will exceed 55%: P(X>55)
Area=
(from Z Table at Z=2.5)
Required Area = =
(Total area is 1)
P(X>55)
Business Statistics:Topic 5

16. Business Statistics:Topic 5
Example
The probability that the return will exceed 22%: P(X>22)
Required Area = =
Area= (from Z Table at Z=.8)
P(X>22)
Area=0.5
Business Statistics:Topic 5

17. Business Statistics:Topic 5
Example
The probability that the return will be between 22% and 55%
Required Area = =
P(22<X<55)
0.2881
0.4938
Business Statistics:Topic 5

18. ‘X’ Value for a given Probability
Find the value of X, given that P(more than X) is 10% = 0.1
=10
10% =0.1
=30
Business Statistics:Topic 5

19. Business Statistics:Topic 5
Testing For Normality
Using Histogram and Polygon to examine the shape: bell shape
Using the measures of central tendencies: values of Mean, Mode & Median are close to each other
Business Statistics:Topic 5

20. Sampling Distribution
Probability Distribution of sample statistics such as
Sample mean
Sample proportion
Business Statistics:Topic 5

21. Sample Mean Distribution
Mean (Unbiasedness)
Standard Deviation
Note: As the sample size increases, the standard
deviation of the sampling distribution decreases
Business Statistics:Topic 5

22. Properties of Sample Mean Distribution
When the population follows a Normal Distribution:
The Sampling Distribution for the mean follows a Normal distribution with
Business Statistics:Topic 5

23. Properties of Sample Mean Distribution
When the population does not follow a Normal Distribution:
The Sampling Distribution for the mean follows a Normal distribution as long as (n ³ 30) with:
Business Statistics:Topic 5

24. Business Statistics:Topic 5
Central Limit Theorem
As sample size gets large enough (³ 30) ...
sampling distribution becomes almost normal.
Business Statistics:Topic 5

25. ‘Z’ Score for Sample Mean Distribution
Sampling Distribution
Standardized Normal Distribution
Business Statistics:Topic 5

26. Business Statistics:Topic 5
Example
The Dean of a business school claims that the average weekly income of graduates of this school three years after graduation is $700.
If the Dean’s claim is correct and if weekly incomes are normally distributed with a standard deviation of $60, what is the probability that 36 randomly selected graduates have an average weekly income of less than $675?
Business Statistics:Topic 5

27. Business Statistics:Topic 5
Example =
0.4938
675
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28. Distribution of Sample Proportion
Approximated as Normal distribution, provided
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29. Sample Proportion Distribution
Mean: Population Proportion P
Standard Deviation:
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30. ‘z’ Score for Sample Proportion ps
Standardized Normal Distribution
Sample Proportion Distribution
Business Statistics:Topic 5

31. Business Statistics:Topic 5
Example
If the population proportion is 0.5, from a
sample of 100, what is the probability that the
sample proportion exceeds 0.7?
Business Statistics:Topic 5

32. Business Statistics:Topic 5
Example
Probability of ps > 0.7 = ? =
P(ps > 0.7)
P(z > 2.82)
= 0.5
Business Statistics:Topic 5

33. Business Statistics:Topic 5
Summary
In this topic you have Discussed:
the characteristics of the normal distribution
problem solving involving the normal distribution
Sampling Distributions: sample mean & sample proportion
Business Statistics:Topic 5