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How do you know when to use a specific distribution? For example, when would we use the Normal, binomial, grometric, hypergeometric, or negative binomial distribution? Could you give a few examples?

Question

How do you know when to use a specific distribution? For example, when would we use the Normal, binomial, grometric, hypergeometric, or negative binomial distribution? Could you give a few examples?

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Step 1

Normal distribution:

Several tests of normality exist, using which you can verify whether a particular data follows the normal distribution.

Usually, before conducting a formal test, we prefer to take the help of graphical methods, to see if the data may be assumed to follow the normal distribution, at least approximately. A few such graphical methods are:

  • Histogram of the data , superimposed with a normal probability curve,
  • Normal probability plot with confidence interval,
  • Normal quantile-quantile (QQ) plot.
  • Boxplot, etc.

If the graphical display appears to show at least an approximate normal distribution, then a formal test can be used to verify the normality. A few such tests are as follows:

  • Pearson’s Chi-squared test for goodness of fit,
  • Shapiro-Wilk test,
  • Kolmogorov-Smirnov test, etc.

Assumptions of normality:

The assumptions of normality are as follows:

  • The data should be symmetric.
  • The data should be mesokurtic.
  • The empirical rule must be satisfied.

Empirical rule:

  • 68% of values fall within one standard deviation from the mean.
  • 95% of values fall within two standard deviations from the mean.
  • 99.7% of values fall within three standard deviations from the mean.
Step 2

Example:

Consider the salaries of the elementary school teacher in a particular city with mean value of 31,000 and a standard deviation of 3,000 dollars.

Let X is the salary of a randomly selected elementary school teacher in the United States. It is mentioned that, µ­ = 31,000 and σ = 3,000.

Here, a sample of n = 100 teachers is considered, which is quite a large sample size. So, it may be assumed that the sample mean has an approximate normal distribution with µ­ = 31,000 and σ/√n = 3,000/100 = 300.

The probability that the mean salary of 100 elementary school teachers is greater than $31,500 can be obtained by standardizing the random variable and using Excel as follows:

 

X-31,500-[by standardizing
P(X>31,500 P
PZ31.500 -3 1,000 [z N(0,1),ie, Z is standard normal variate
300
P(Z 1.667)
1-P(Z<1.667)
1-0.9522
Excel formula: =NORM.S.DIST(1.667,1)
0.0478
help_outline

Image Transcriptionclose

X-31,500-[by standardizing P(X>31,500 P PZ31.500 -3 1,000 [z N(0,1),ie, Z is standard normal variate 300 P(Z 1.667) 1-P(Z<1.667) 1-0.9522 Excel formula: =NORM.S.DIST(1.667,1) 0.0478

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Step 3

Binomial distribution:

The binomial distribution gives the probability of number of successes out of n trials in a series of Bernoulli trial...

piqx0,1. n;0 < p<1;q 1-p
P(x)
otherwise
E(x) np
(х)- пр(1 - р)
help_outline

Image Transcriptionclose

piqx0,1. n;0 < p<1;q 1-p P(x) otherwise E(x) np (х)- пр(1 - р)

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