Topic 6 - Normal Distribution and Estimation

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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

[1] 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

[2]

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

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,.

  1. Textbook Pg. 270-287
  2. Textbook Pg. 335-353
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