6. Normal Distribution and Continuous Random Variables

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?

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?

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?

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

Which of the following statements is true about a z-score?

Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Using a standard normal distribution, what is the $z$-score such that the area to the left of $z$ is $0.52$?

Government Solutions?In May, 2000, the Gallup Organization reported that 11% of adult Americans had a great deal of trust and confidence in the federal government handling domestic issues. Suppose a survey of a random sample of 1100 adult Americans finds that 84 have a great deal of trust and confidence in the federal government handling domestic issues. Would these results be considered unusual? Why?

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

Water Footprint A water footprint is a measure of the appropriation of fresh water. The water footprint (in cubic meters) for a kilogram of wheat can be approximated by a normal distribution, as shown in the figure. (Source: Ecological Indicators)

b. What water footprint represents the 29th percentile?

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 (c) at least 10. Identify any unusual events. Explain.

Sketch a graph to represent the probability, then use a calculator to find it.

$P\left(Z>1.14\right)$

Use a calculator to find the z–scores of the region shown in the standard normal distribution below.

A = 0.800

Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.

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Greater than 3.00 minutes

Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.

Less than 4.00 minutes

Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.

Between 2 minutes and 3 minutes