Question

11. Answer true or false to each statement. Explain your answers.

a. Two normal distributions that have the same mean are centered at the same place, regardless of the relationship between their standard deviations.

b. Two normal distributions that have the same standard deviation have the same spread, regardless of the relationship between their means.

Short answer

a. The statement of two normal distributions that have the same mean is centered at the same place, regardless of the relationship between their standard deviations is a true statement.

b. The statement of two normal distributions that have the same standard deviation have the same spread, regardless of the relationship between their means is a true statement.

Step-by-step solution

Part (a) Step 1: Concept Introduction

A standard score, often known as a $z-$score, is the normal random variable of a standard normal distribution. The following equation can be used to convert any normal random variable $X$into a $z$score:

$Z=\frac{{\displaystyle (X-\mu )}}{\sigma }$

Part (a) Step 2: Explanation 

Think of "two normal distributions with the same mean-centered in the same spot, regardless of their standard deviation relationships."

As shown in the diagram below, two variables with the same mean and standard deviation are distributed differently.

So, the statement given is true.

Part (b) Step 3: Concept Introduction

A standard score, often known as a $z-$score, is the normal random variable of a standard normal distribution. The following equation can be used to convert any normal random variable $X$into a

$z$score:

$Z=\frac{(X-\mu )}{\sigma }$

Part (b) Step 4: Explanation 

In spite of the relationship between their means, two normal distributions with the same standard deviation will have the same spread.

As shown in the below figure, the two variables have the same mean and same standard deviation but different distributions.

So, the statement given is true.