Why is it correct to say a normal distribution and the standard normal distribution?

A normal distribution is a general distribution that represents any normally distributed data with any possible value for its parameters, that is, the mean and the standard deviation. Conversely, the standard normal distribution is a specific case where the mean equals zero and the standard deviation is the unit. That is why we can refer to a normal distribution and the standard normal distribution.

Step-by-step explanation:

We have to remember that a normal distribution has two parameters that define it, namely, the mean and the standard deviation, and there are, theoretically, infinite possible means and standard deviations, so we tell about a normal distribution in general.

Conversely, the standard normal distribution is a normal distribution with a mean = 0 and a standard deviation = 1, and we also have to remember that is possible to 'convert' or 'transform' any raw score from any normally  distributed data into a z-score to find its probability using the standard normal distribution. The formula for a z-score is as follows:

Where

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In other words, the standard normal distribution is a specific case for normally distributed data whose values are standardized or represent distances from the mean in standard deviations units, and thanks to this, we can find any associated probability with these values for any possible normal distribution (see the graph below).

try to find the inequality

Step-by-step explanation:

and the vertex is 5