6. Normal Distribution and Continuous Random Variables

The random variable x is normally distributed with the given parameters. Find each probability.

a. μ = 9.2, σ ≈ 1.62, P(x < 5.97)

Significance For bone density scores that are normally distributed with a mean of 0 and a standard deviation of 1, find the percentage of scores that are

c. not significant (or less than 2 standard deviations away from the mean).

In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.

P(72 < x < 82)

Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.

Bone Density Find the bone density score that is the 90th percentile, which is the score separating the lowest 90% from the top 10%.

Graphical Analysis In Exercises 19–22, use the box-and-whisker plot to determine whether the shape of the distribution represented is symmetric, skewed left, skewed right, or none of these. Justify your answer.

Hershey Kisses Based on Data Set 38 “Candies” in Appendix B, weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g

b. What is the value of the median?

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

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

Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.

z = -0.51

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Advanced Dental Admission Test The Advanced Dental Admission Test (ADAT) is designed so that the scores fit a normal distribution, as shown in the figure. (Source: American Dental Association)

b. Between what two values does the middle 50% of the ADAT scores lie?

Sleepwalking Assume that 29.2% of people have sleepwalked (based on “Prevalence and Comorbidity of Nocturnal Wandering in the U.S. Adult General Population, by Ohayon et al., Neurology, Vol. 78, No. 20). Assume that in a random sample of 1480 adults, 455 have sleepwalked.

a. Assuming that the rate of 29.2% is correct, find the probability that 455 or more of the 1480 adults have sleepwalked.

Critical Values. In Exercises 41–44, find the indicated critical value. Round results to two decimal places.

z0.90

Sleepwalking Assume that 29.2% of people have sleepwalked (based on “Prevalence and Comorbidity of Nocturnal Wandering in the U.S. Adult General Population, by Ohayon et al., Neurology, Vol. 78, No. 20). Assume that in a random sample of 1480 adults, 455 have sleepwalked.

c. What does the result suggest about the rate of 29.2%?

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z = -1.95

Normal Distribution Using a larger data set than the one given for the preceding exercises, assume that cell phone radiation amounts are normally distributed with a mean of 1.17 W/kg and a standard deviation of 0.29 W/kg.

b. Find the value of Q3, the cell phone radiation amount that is the third quartile.