6. Normal Distribution and Continuous Random Variables

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)

SAT Total Scores Use the normal distribution in Exercise 13.

b. Out of 1000 randomly selected SAT total scores, about how many would you expect to be greater than 1100?

Pregnancy Length Use the normal distribution in Exercise 15.

a. What percent of the new mothers had a pregnancy length of less than 290 days?

Red Blood Cell Count Use the normal distribution in Exercise 16.

a. What percent of the adult males have a red blood cell count less than 6 million cells per microliter?

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

Weights of Teenagers In a survey of 18-year old males, the mean weight was 166.7 pounds with a standard deviation of 49.3 pounds. (Adapted from National Center for Health Statistics)

c. What weight represents the first quartile?

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

COVID-19 Response Surveyors asked respondents to rate ten key aspects of their government’s response to the COVID-19 pandemic, including preparedness, communication, and material aid. A pandemic response score that ranged from 0 to 100 was calculated. The mean score for U.S. respondents was 50.6 with a standard deviation of 29.0. (Source: PLOS One)

b. What score represents the 61st percentile?

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

Billboard Hot 100 The length (in seconds) of the 100 most popular songs during the week of May 5, 2021, can be approximated by a normal distribution, as shown in the figure. (Source: Spotify)

b. What song length represents the 17th percentile?

In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z = 0.72

In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

Between z = 0 and z = 2.95

The life spans of car batteries are normally distributed, with a mean of 44 months and a standard deviation of 5 months.

c. What is the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies?

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)

What is the highest score that would still place a person in the bottom 10% of the scores?

Find each probability using the standard normal distribution.

b. P(z < 2.23)

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.