Questions: Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample. If two normal distributions have the same mean and standard deviation, then they have the same shape. Choose the correct answer below. A. The statement is false because the normal distribution with a mean equal to -1 and a standard deviation equal to 2 can change shape based on the value of x. B. The statement is true because a normal distribution is completely defined by its mean and standard deviation. C. The statement is true because all normal distributions have the same shape. D. The statement is false because the normal distribution with a mean equal to 1 and a standard deviation equal to 2 can change shape based on the value of x.

Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample.
If two normal distributions have the same mean and standard deviation, then they have the same shape.

Choose the correct answer below.
A. The statement is false because the normal distribution with a mean equal to -1 and a standard deviation equal to 2 can change shape based on the value of x.
B. The statement is true because a normal distribution is completely defined by its mean and standard deviation.
C. The statement is true because all normal distributions have the same shape.
D. The statement is false because the normal distribution with a mean equal to 1 and a standard deviation equal to 2 can change shape based on the value of x.

Transcript text: Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample. If two normal distributions have the same mean and standard deviation, then they have the same shape. Choose the correct answer below. A. The statement is false because the normal distribution with a mean equal to -1 and a standard deviation equal to 2 can change shape based on the value of $x$. B. The statement is true because a normal distribution is completely defined by its mean and standard deviation. C. The statement is true because all normal distributions have the same shape. 7. The statement is false because the normal distribution with a mean equal to 1 and a standard deviation equal to 2 can change shape based on the value of $x$.

Solution

Solution Steps

To determine the validity of the statement, we need to understand the properties of normal distributions. A normal distribution is completely characterized by its mean and standard deviation. If two normal distributions have the same mean and standard deviation, they are identical in shape, as these parameters define the distribution's center and spread. Therefore, the statement is true.

Step 1: Understand the Properties of Normal Distributions

A normal distribution is defined by its mean $\mu$ and standard deviation $\sigma$. The shape of a normal distribution is completely determined by these two parameters. If two normal distributions have the same $\mu$ and $\sigma$, they are identical in shape.

Step 2: Analyze the Given Statement

The statement claims that if two normal distributions have the same mean and standard deviation, then they have the same shape. Given the properties of normal distributions, this statement is true because the mean and standard deviation uniquely define the distribution's shape.

Step 3: Evaluate the Multiple-Choice Options

Option A: Incorrect. The shape of a normal distribution does not change based on the value of $x$ if the mean and standard deviation are fixed.
Option B: Correct. A normal distribution is completely defined by its mean and standard deviation.
Option C: Incorrect. While all normal distributions have a bell shape, their specific shape is determined by their mean and standard deviation.
Option D: Incorrect. Similar to Option A, the shape does not change with $x$ if the mean and standard deviation are constant.

Final Answer

The correct answer is $\boxed{\text{B}}$.

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