Final answer:
The term "a" normal distribution refers to any normal distribution with specific mean (μ) and standard deviation (σ), while "the" standard normal distribution refers to the specific case with a mean of 0 and standard deviation of 1, used for standardizing scores across differently scaled data sets.
Step-by-step explanation:
It is correct to say "a" normal distribution and "the" standard normal distribution because the terms refer to different concepts within probability and statistics. A normal distribution can have any mean (μ) and standard deviation (σ), which are its parameters, and is denoted as X~N(μ, σ). These distributions are depicted as bell-shaped curves and are prevalent in various disciplines. When we refer to any normal distribution with a particular mean and standard deviation, we can say "a" normal distribution to imply that it is one instance of many possible distributions.
The standard normal distribution, on the other hand, is a specific case of the normal distribution with a mean of 0 and standard deviation of 1. It is represented using z-scores (standardized scores), allowing for comparison across different sets of data. Because there is only one such distribution, it is appropriate to refer to it as "the" standard normal distribution, denoted as Z~N(0, 1). This unique distribution has ubiquitous applications, such as determining probabilities and comparing scores across different samples.
It is incorrect to refer to any distribution that is not normal as the standard normal distribution. Only those distributions that have the specific parameters of a mean of 0 and standard deviation of 1 should be referred to as "the" standard normal distribution, while other normal distributions should be referred to as "a" normal distribution.