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# 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.**