Question

- What is a normal distribution and why is it useful for understanding probabilities and statistics? Provide an example of a normal distribution and explain why it can be categorized as a normal distribution.

- What is the difference between a standard normal distribution and a normal distribution?

- How do we standardize a normal distribution?

- What is a z-score? How do we calculate the z-score for a normal variable?

- How could the standard normal (z) distribution table be used to calculate the probability of:

a) the cumulative area less than a z-score?

b) the cumulative area greater than a z-score?

c) the area between two z-scores?

Directions:  you are required to “Start a New Thread” by Day 3. Initial post should be 200-300 words

Answer 1

A normal distribution which appears as a bell curve is a probability distribution that is symmetric about the mean. Here mean is zero and the standard deviation is 1.

It is useful in probabilities and statistics because it can be applied in natural phenomena.

An example is the Height of girls, here normal distribution is used because small differences between a person's height and the mean occur more frequently than substantial deviations from the mean.

The normal distribution is called Standard Normal distribution if mean is 0 and the standard deviation is 1.

Change in a normal distribution to standardized distribution by applying this transformation to (X) variable

Z=(X - mean (X)) /SD(X).

A z score is a raw score in the context of distance from the mean when measured in standard deviation.

The z score is positive if the value is above the mean, and negative if it is below the mean.

Formula to calculate z score is z = (x-μ)/σ

where sigma is the standard deviation

μ is mean

x is the raw score.

First, we find a z score with the formula mentioned, then with the help of the standard normal distribution table, we find the cumulative probability associated with the z-score.

To find the area under a normal curve, find the z-score of the data value and use a Z-Score table to find the area. A Z-Score Table is a table that shows the area percentage to the left of a given z-score on a standard normal distribution.

we can find P(Z < a) or P(Z > a) from the table and another thing we can find by this formula,

P(Z > a) = 1 - P(Z < a)., if we find P(Z < a) and vice versa.

To find the area between two z scores we can use this formula, P(a < Z < b) = P(Z < b) - P(Z < a)..