6. Normal Distribution and Continuous Random Variables

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=1.365

For the standard normal distribution, what proportion of the distribution lies between $z=-1.75$ and $z=1.75$?

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

A random variable Z that is normally distributed with mean μ = 0 and standard deviation σ = 1 is said to have the ____________________.

For the standard normal probability distribution, what are the mean $\mu$ and standard deviation $\sigma$?

Student's $t$ distributions are symmetric about a value of $t$. What is that value?

Find the area under the standard normal distribution to the left of a z-score of $1.21$.

Find the area under the standard normal distribution to the right of a z-score of $-0.44$.

Find the area of the shaded region under the standard normal distribution.

Find the z-score such that $P$(Z < z)$=0.6331$.

Find the z-score corresponding to the probability/area shown under the standard normal curve below.

Normal Distribution If the following data are randomly selected, which are expected to have a normal distribution?

a. Weights of Reese’s Peanut Butter Cups

Standard Normal Distribution. In Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.

Between 1.50 and 2.00

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):

Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

If the Navy changes the height requirements so that all women are eligible except the shortest 3% and the tallest 3%, what are the new height requirements for women?

Normal Distribution If the following data are randomly selected, which are expected to have a normal distribution?

d. Exact volumes of Coke in 12 oz cans