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=0.33

Seat Designs. In Exercises 7–9, assume that when seated, adult males have back-to-knee lengths that are normally distributed with a mean of 23.5 in. and a standard deviation of 1.1 in. (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats.

Find the probability that nine males have back-to-knee lengths with a mean greater than 23.0 in.

Which of the following is a similarity between the $t$ distribution and the standard $z$ distribution?

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

P85

Bone Density Test A bone mineral density test is used to identify a bone disease. The result of a bone density test is commonly measured as a z score, and the population of z scores is normally distributed with a mean of 0 and a standard deviation of 1.

e. If the mean bone density test score is found for 9 randomly selected subjects, find the probability that the mean is greater than 0.23.

Continuity Correction In testing the assumption that the probability of a baby boy is 0.512, a geneticist obtains a random sample of 1000 births and finds that 502 of them are boys. Using the continuity correction, describe the area under the graph of a normal distribution corresponding to the following. (For example, the area corresponding to “the probability of at least 502 boys” is this: the area to the right of 501.5.)

c. The probability of more than 502 boys

Given the mean of a normal distribution is $\mu$ and the standard deviation is $\sigma$, what is the mean of the standard normal distribution after standardizing the variable?

In a survey of U.S. adults, 81% feel they have little or no control over data collected about them by companies. You randomly select 250 U.S. adults and ask them whether they feel they have control over data collected about them by companies. Use this information in Exercises 11 and 12. (Source: Pew Research Center)

Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

Which of the following statements is true about the standard normal distribution?

If an observation has a z-score of $z=0$ in the standard normal distribution, what does this indicate about the observation's value relative to the mean?

If a random variable is normally distributed with a mean of $0$ and a standard deviation of $1$, what is this distribution called?

Computing Probabilities for Normal Distributions In Exercises 1–6, the random variable x is normally distributed with mean mu=174 and standard deviation sigma=20. Find the indicated probability.

P(172 < x <192)

For the standard normal distribution, what is the area under the curve between $z=0.0$ and $z=1.79$?

For a standard normal distribution, what is the probability that $z>1.75$?

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