6. Normal Distribution and Continuous Random Variables

Which of the following does not describe the standard normal distribution?

Interpreting Normal Quantile Plots Which of the following normal quantile plots appear to represent data from a population having a normal distribution? Explain.

Which of the following is most likely to be modeled by a $standardnormaldistribution$?

Which of the following does not describe the standard normal distribution?

For a normal distribution, which of the following statements is true about the $\mathrm{mean}$, $\mathrm{median}$, and $\mathrm{mode}$?

For the standard normal distribution, what is the total area under the normal curve?

Which of the following is a characteristic of the standard normal probability distribution?

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.

Safe Loading of Elevators The elevator in the car rental building at San Francisco International Airport has a placard stating that the maximum capacity is “4000 lb—27 passengers.” Because 4000/27=148, this converts to a mean passenger weight of 148 lb when the elevator is full. We will assume a worst-case scenario in which the elevator is filled with 27 adult males. Based on Data Set 1 “Body Data” in Appendix B, assume that adult males have weights that are normally distributed with a mean of 189 lb and a standard deviation of 39 lb.

b. Find the probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.

In a standard normal distribution, what is the mean $\mu$ of the distribution?

Which of the following is a property of the standard normal distribution?

Using the Central Limit Theorem. In Exercises 5–8, assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 1.2 kg and a standard deviation of 4.9 kg (based on Data Set 13 “Freshman 15” in Appendix B).

a. If 1 male college student is randomly selected, find the probability that he gains between 0.5 kg and 2.5 kg during freshman year.

For a standard normal distribution, what is the mean $\mu$ and standard deviation $\sigma$?

Uniform Distribution A uniform distribution is a continuous probability distribution for a random variable x between two values a and b (a<b), where (a ≤ x ≤ b) and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown below.

The probability density function of a uniform distribution is

on the interval from (x=a) to (x=b). For any value of x less than a or greater than b, y=0 . In Exercises 59 and 60, use this information.

For two values c and d, where a ≤ c < d ≤ b, the probability that x lies between c and d is equal to the area under the curve between c and d, as shown below.

So, the area of the red region equals the probability that x lies between c and d. For a uniform distribution from (a=1) to (b=25) , find the probability that

d. x lies between 8 and 14.

Finding Area

In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z=0.33