# Topic 6 - Normal Distribution and Estimation

## Contents

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

### Normal Distribution Functions

 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

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

### 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

#### 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

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)

## Introduction To Estimation

### 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)
• Interval Estimator
• Draws inferences about a population by estimating the value of an unknown parameter using an interval

### 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

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