Chapter 7 The Normal Probability Distribution

Question 1

Properties of the Normal Curve

Answer 1

1. It is symmetric about its mean, μ.
2. Because mean = median = mode, the curve has a single peak and the highest point occurs at x = μ.
3. The area under the curve is 1.
4. Because the curve is perfectly bell-shaped and symmetric, the area under the curve to the right of μ equals the area under the curve to the left of μ, which equals 1/2.
5. As x increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis (i.e., it is asymptotic to the positive x-axis). As x decreases without bound (gets more and more negative), the graph approaches, but never reaches, the horizontal axis (i.e., it is asymptotic to the negative x-axis).
6. Because it is perfectly bell-shaped, the Empirical Rule is applicable:
   ∙ Approximately 68% of the area under the normal curve is between x = μ – σ and x = μ + σ;
   ∙ Approximately 95% of the area is between x = μ – 2σ and x = μ + 2σ;
   ∙ Approximately 99.7% of the area is between x = μ – 3σ and x = μ + 3σ.

Question 2

Why is the mean (μ) referred to as the location parameter?

Answer 2

While keeping the standard deviation (σ) fixed, changing the value of μ has the effect of “shifting” the graph to the left or right.

Question 3

Why is the standard deviation (σ) referred to as the shape parameter?

Answer 3

While keeping the mean (μ) fixed, changing the value of σ has the effect of “flattening or squeezing” the graph in the vertical direction.

Question 4

The Standard Normal Distribution

Answer 4

A normal distribution with the special values of mean μ = 0 and standard deviation σ = 1.

To distinguish this very special normal curve from any other normal curve, the horizontal axis is labeled “Z”, (as opposed to “X” , which we use for all other normal curves).

Question 5

What are the Two Ways of Interpreting the Area under a Normal Curve?

Answer 5

1. As the probability that a randomly selected member from the population is lies within the interval that defines the area
2. As the proportion of the population that lies within the interval that defines the area