Continuous random variables and Normal distributions

Chapter focus is on:

v Continuous random variables

v Normal distributions

This section presents the standard normal distribution which has three properties:

1. It is bell-shaped.

2. It has a mean equal to 0.

3. It has a standard deviation equal to 1.

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It is extremely important to develop the skill to find areas (or probabilities or relative frequencies) corresponding to various regions under the graph of the standard normal distribution.

This section presents the standard normal distribution which has three properties:

1. It is bell-shaped.

2. It has a mean equal to 0.

3. It has a standard deviation equal to 1.

●

It is extremely important to develop the skill to find areas (or probabilities or relative frequencies) corresponding to various regions under the graph of the standard normal distribution.

A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:

1. The total area under the curve must equal 1.

2. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.)

Because the total area under the density curve is equal to 1,

there is a correspondence between area and probability

v The standard normal distribution is a probability distribution with mean equal to 0 and standard deviation equal to 1, and the total area under its density curve is equal to 1.

Normal Distribution

vThe mean is located on the centerline. Scores lower than the mean will have a negative Z-score, while scores higher than the mean will have positive Z-scores.

Example – Weights of Water Taxi Passengers

In the Chapter Problem, we noted that the safe load for a water taxi was found to be 3500 pounds. We also noted that the mean weight of a passenger was assumed to be 140 pounds. Assume that the weights of the men are normally distributed with a mean of 172 pounds and standard deviation of 29 pounds. If one man is randomly selected, what is the probability he weighs less than 174 pounds?

1. Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the x value(s) being sought.

2. Use Table A-2 to find the z score corresponding to the cumulative left area bounded by x. Refer to the body of Table A-2 to find the closest area, then identify the corresponding z score.

3. Using Formula 6-2, enter the values for µ, s, and the z score found in step 2, then solve for x.

x = µ + (z • s) (Another form of Formula 6-2)

(If z is located to the left of the mean, be sure that it is a negative number.)

4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem.