6. Normal Distribution and Continuous Random Variables

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

c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)

Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.

Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.

Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features.

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?

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

Red Blood Cell Count The red blood cell counts (in millions of cells per microliter) for a population of adult males can be approximated by a normal distribution, with a mean of 5.4 million cells per microliter and a standard deviation of 0.4 million cells per microliter.

a. What is the minimum red blood cell count that can be in the top 25% of counts?

Bags of Baby Carrots The weights of bags of baby carrots are normally distributed, with a mean of 32 ounces and a standard deviation of 0.36 ounce. Bags in the upper 4.5% are too heavy and must be repackaged. What is the most a bag of baby carrots can weigh and not need to be repackaged?

History Grades In a history class, the grades for various assessments are all positive numbers and have different distributions. Determine whether the grades for each assessment could be normally distributed. Explain your reasoning.

b. a final with a mean of 72, standard deviation of 9, and 90th percentile score of 93

History Grades In a history class, the grades for various assessments are all positive numbers and have different distributions. Determine whether the grades for each assessment could be normally distributed. Explain your reasoning.

e. an extra credit assignment with a mean of 2.25 and a standard deviation of 2.49

Ice Cream The weights of ice cream cartons are normally distributed with a mean weight of 10 ounces and a standard deviation of 0.5 ounce.

a. What is the probability that a randomly selected carton has a weight greater than 10.21 ounces?

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(x < 55)

In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(0.42 < z < 3.15)

In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(z < 1.28)

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 = -1.5 and to the right of z = 1.5

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

Between z = -1.55 and z = 1.04