6. Normal Distribution and Continuous Random Variables

In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.

n=24, p=0.85, q=0.15

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

Doorway Height The Boeing 757-200 ER airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. (based on Data Set 1 “Body Data” in Appendix B).

a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.

In Exercises 8 and 9, assume that women have standing eye heights that are normally distributed with a mean of 59.7 in. and a standard deviation of 2.5 in. (based on anthropometric survey data from Gordon, Churchill, et al.).

a. If an eye recognition security system is positioned at a height that is uncomfortable for women with standing eye heights less than 54 in., what percentage of women will find that height uncomfortable?

Which of the following is true regarding the standard normal distribution?

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

Assume the random variable $x$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. What is the standard normal variable $z$ corresponding to a value $x$?

For the standard normal probability distribution, the area to the left of the $\mathrm{mean}$ is:

In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

 0.1

For the standard normal distribution, what $z$-value corresponds to a lower-tail probability of $1\%$?

For a two-tailed hypothesis test at the $5\%$ significance level using the standard normal distribution, what are the critical value(s) for $z$?

For a standard normal distribution, if the probability that a value is less than $z$ is $0.975$, what is the smallest value of $z$ that satisfies this requirement?

In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z = 0.72

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

For the standard normal distribution ($Z\sim N(0,1)$), what is the area under the curve to the right of the mean ($\mu =0$)?

In a perfectly normal distribution of scores, what percentage of the data falls within $1\sigma$ of the mean $\mu$?