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 z-score that has 78.5% of the distribution’s area to its left.

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

Find the z-score that has 2.275% of the distribution’s area to its left.

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.

In Exercises 33 and 34, find the indicated probabilities. If convenient, use technology to find the probabilities.

The daily surface concentration of carbonyl sulfide on the Indian Ocean is normally distributed, with a mean of 9.1 picomoles per liter and a standard deviation of 3.5 picomoles per liter. Find the probability that on a randomly selected day, the surface concentration of carbonyl sulfide on the Indian Ocean is

a. between 5.1 and 15.7 picomoles per liter.

Find the positive z-score for which 94% of the distribution’s area lies between -z and z.

What braking distance represents the first quartile?

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)

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

Find the value of x that has 88.3% of the distribution’s area to its left.

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

MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (b) between 490 and 510. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)

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

MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (c) more than 515. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)

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 (b) between 20 and 28 minutes.

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.

In Exercises 5 and 6, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.

A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (a) exactly 7. Identify any unusual events. Explain.

In Exercises 5 and 6, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.

A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (b) less than 5. Identify any unusual events. Explain.

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