6. Normal Distribution and Continuous Random Variables

Given the mean of a normal distribution is $\mu$ and the standard deviation is $\sigma$, what is the mean of the corresponding standard normal distribution after standardizing the variable?

For the standard normal distribution, what is the area to the right of $1$ (that is, $P(Z>1)$)?

Using the standard normal distribution, what is the value of $P(0<z<2.25)$?

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(- 1.54 < z < 1.54)

Suppose the random variable $Z$ follows a standard normal distribution. What is the area under the standard normal curve to the left of $Z=1$?

In a standard normal distribution, what percentage of the data falls within one standard deviation of the mean (that is, between $-1$ and $1$ on the $z$-score scale)?

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≤ 150)

Basis for the Range Rule of Thumb and the Empirical Rule. In Exercises 45–48, find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.

About __ % of the area is between z = -3.5 and z = 3.5 (or within 3.5 standard deviation of the mean).

In the $standardnormal$ distribution, what happens to the graph of the normal curve as the mean $\mu$ increases while the standard deviation $\sigma$ remains constant?

Employee Wellness A survey of employed U.S. adults found that only 35% believe their employer cares about their well-being. You randomly select a sample of U.S. employees. Find the probability that fewer than 100 believe their employer cares about their well-being. (Source: Gallup)

c. You select 400 U.S. employees.

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.

Safe Loading of Elevators The elevator in the car rental building at San Francisco International Airport has a placard stating that the maximum capacity is “4000 lb—27 passengers.” Because 4000/27=148, this converts to a mean passenger weight of 148 lb when the elevator is full. We will assume a worst-case scenario in which the elevator is filled with 27 adult males. Based on Data Set 1 “Body Data” in Appendix B, assume that adult males have weights that are normally distributed with a mean of 189 lb and a standard deviation of 39 lb.

a. Find the probability that 1 randomly selected adult male has a weight greater than 148 lb.

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

b. μ = 87, σ ≈ 19, P(x > 40.5)

"According to data from the city of Toronto, Ontario, Canada, there were nearly 112,000 parking infractions in the city for December 2020, with fines totaling over 5,500,000 Canadian dollars. The fines (in Canadian dollars) for a random sample of 105 parking infractions in Toronto, Ontario, Canada, for December 2020 are listed below. (Source: City of Toronto)

In Exercises 1–5, use technology. If possible, print your results.

Draw a histogram for the data. Does the distribution appear to be bell-shaped?"

Find the area under the standard normal distribution to the right of a z-score of $-0.44$.

Assume $x$ is normally distributed with a mean of $10$ and a standard deviation of $2$. What is the z-score for $x=14$?