6. Normal Distribution and Continuous Random Variables

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.

Find the positive z-score for which 12% of the distribution’s area lies between and z.

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.

Health Club Schedule The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (c) more than 30 minutes.

Suppose a normal distribution is shown with a mean of $\mu =0$ and a standard deviation of $\sigma =1$. If the shaded area to the left of the indicated point is $0.8413$, what is the z-score corresponding to the indicated point?

In Problems 5–12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found.

Determine the area under the standard normal curve that lies to the left of

a. z = –2.45

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

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Given a normal distribution with mean $\mu$ = $0$ and standard deviation $\sigma$ = $1$, what are the z-scores that correspond to the boundaries of the middle 95% of the distribution as shown in a standard normal curve graph?

Computing and Interpreting z-Scores In Exercises 39 and 40, (a) find the z-score that corresponds to each value and (b) determine whether any of the values are unusual.

Stanford-Binet IQ Scores The test scores for the Stanford-Binet Intelligence Scale are normally distributed with a mean score of 100 and a standard deviation of 16. The test scores of four students selected at random are 98, 65, 106, and 124.

Suppose a normal distribution is shown on a graph with the mean marked at the center. If a point is marked one standard deviation above the mean, which value best represents the $z$-score of that point?

In Problems 5–12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found.

Determine the area under the standard normal curve that lies to the left of

d. z = 3.49

Explain how to transform a given x-value of a normally distributed variable x into a z-score.

In Exercises 2–4, the random variable x is normally distributed with mean mu= 18 and standard deviation sigma 7.6

Find each probability.

b. P(0 < x < 5)

Converting observations into $z$-scores is also called standardizing the observations.

z Scores. In Exercises 5–8, express all z scores with two decimal places.

Diastolic Blood Pressure of Females For the diastolic blood pressure measurements of females listed in Data Set 1 “Body Data” in Appendix B, the highest measurement is 98 mm Hg. The 147 diastolic blood pressure measurements of females have a mean of 70.2 mm Hg and a standard deviation of 11.2 mm Hg.

c. Convert the highest diastolic blood pressure to a z score.

Comparing z-Scores from Different Data Sets The table shows population statistics for the ages of Best Actor and Best Supporting Actor winners at the Academy Awards from 1929 to 2020. The distributions of the ages are approximately bell-shaped. In Exercises 51–54, compare the z-scores for the actors.

Best Actor 1970: John Wayne, Age: 62

Best Supporting Actor 1970: Gig Young, Age: 56

True or False? In Exercises 7–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

It is impossible to have a z-score of 0.