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?

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:

Step 3

**Binomial distribution:**

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

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