# Topic 6 - Normal Distribution and Estimation

This article is a topic within the subject Business & Economic Statistics.

## Contents |

## Required Reading

Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition, pp. 270-287 & 335-353.

### Normal Distribution Functions

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The **Normal Distribution** is **symmetrical, uni-modal and is bell curve shaped**. In this distribution, the **mean = median = mode**.
The 2 factors that influence the shape of the bell curve are the mean (the middle number) and the variance (how steep/flat)
Consequentially, we are able to define a normal distribution for the random variable 'X' in the following way:

**X ~ (μ,σ^2)** where -∞≤x≤∞
This means 'X', the random variable is normally distributed with mean 'μ' and variance 'σ^2'

Statisticians use Z scores to standardise data values so that we can more easily find probability values through a Z standard normal distribution (this link contains examples).

**Z Score = (x-μ)/σ (Linear Function)**

The normal distribution for Z scores (standard normal distribution) is defined in the following way **Z~N(0,1)**. Once we have standardised scores for the standard normal distribution we can use the standard normal table to find probability values for a random variable being a range of values e.g. if the random variable, 'X', is equal to the weight of a person in kilograms, we could use this method to find the probability of a persons weight being between 50kg. and 65kg.

#### Normal Distribution Example

**X ~ (50,100)**

- Find:
**P(45 ≤ x ≤ 60)** **Standardise Scores!!!**- (45-50)/10 ≤ Z ≤ (60-50)/10
- = -1/2 ≤ Z ≤ 1
- = 0 ≤ Z ≤ 1/2 + 0 ≤ Z ≤ 1 (By Symmetry - Because P(-1/2 ≤ Z ≤ 0) = P(0 ≤ Z ≤ 1/2)). We do this because it is much easier to find the probability values in the standard normal table in this format
- = 0.3413 + 0.1915 = 0.5328 (Values from the standard normal table)

- = 0 ≤ Z ≤ 1/2 + 0 ≤ Z ≤ 1 (By Symmetry - Because P(-1/2 ≤ Z ≤ 0) = P(0 ≤ Z ≤ 1/2)). We do this because it is much easier to find the probability values in the standard normal table in this format

- = -1/2 ≤ Z ≤ 1

- (45-50)/10 ≤ Z ≤ (60-50)/10

- Find:

**X ~ (50,100)**

- Find Z_(0.025) and unstandardise it to find the corresponding 'X' value =
- = 1.96
**Unstandardise**!!!!- (1.96 = X – 50 / 10)
- (X = 50 + 19.6) = 69.6

- (1.96 = X – 50 / 10)

- = 1.96

- Find Z_(0.025) and unstandardise it to find the corresponding 'X' value =

### Finding the Value of Z Given the Probabilities

For these questions, we must look at the probabilities within the standard normal table to find the critical z value.

- ZA = 100(1-A)^th Percentile
- Find The 97 ½ Percentile
- P (Z > ZA) = A
- Z_(0.025) (top 2.5%)
- P (Z > Z_0.025) = 0.025
- 1 - 0.025 = 0.9750
- = 1.96

- 1 - 0.025 = 0.9750

- P (Z > Z_0.025) = 0.025

#### Other Questions

Find the value of a standard normal RV where the probability that the RV is GREATER than it is 5%

- 95th Percentile
- = 1.645

- 95th Percentile

Find the Value of a standard normal RV where the probability that the RV is LESS than it is 5%

- 5th Percentile
- = -1.645 (Symmetric)

- 5th Percentile

## Introduction To Estimation

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### Concepts of Estimation

Estimations objective is determine the approximate value of the parameter (e.g. sample mean for population mean. There are 2 types of estimators defined below:

**Point Estimator**- Draws inferences about a population by estimating the value of an unknown parameter using a
**single value**(this will be wrong since a probability of a point on a continuous random variables probability density function is virtually 0)

- Draws inferences about a population by estimating the value of an unknown parameter using a

**Interval Estimator**- Draws inferences about a population by estimating the value of an unknown parameter using an
**interval**

- Draws inferences about a population by estimating the value of an unknown parameter using an

### Characteristics of Good Estimators

**Unbiased**- E(Estimator) = Parameter (e.g. E(sample mean) = population mean)**Consistency**- Difference between the estimator & parameters becomes smaller as sample size increases**Relative Efficiency**- If 2 unbiased estimators of a parameter, the one who’s variance is lower has relative efficiency

## End

This is the end of this topic. Click here to go back to the main subject page for Business and Economic Statistics.

## References

**Textbook** refers to Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition,.