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Chapter 6: Problem 17

Find the area under the standard normal curve a. to the right of \(z=1.36\) b. to the left of \(z=-1.97\) c. to the right of \(z=-2.05\) d. to the left of \(z=1.76\)

Short Answer

Expert verified
a. The area to the right of \(z=1.36\) is 0.0869. b. The area to the left of \(z=-1.97\) is 0.0244. c. The area to the right of \(z=-2.05\) is 0.9798. d. The area to the left of \(z=1.76\) is 0.9608.

Step by step solution

01

Understanding Z-Scores and The Standard Normal Curve

The z-score indicates the number of standard deviations an element is from the mean. A positive z-score indicates a value above the mean, and a negative z-score signifies a value below the mean. To find the area under the curve, we will convert the z-score to a percentile using a Z-table.
02

Find Area to the Right of \(z=1.36\)

First, find the area to the left of \(z=1.36\) using a Z-table. (This should be 0.9131). Since the total area under the curve is 1, subtract this value from 1 to find the area to the right. So, the area to the right of \(z=1.36\) = 1 - 0.9131 = 0.0869.
03

Find Area to the Left of \(z=-1.97\)

Using the Z-table, the area to the left of \(z=-1.97\) is 0.0244.
04

Find Area to the Right of \(z=-2.05\)

First, find the area to the left of \(z=-2.05\) using a Z-table. (This should be 0.0202). Since the total area under the curve is 1, subtract this value from 1 to find the area to the right. So, the area to the right of \(z=-2.05\) = 1 - 0.0202 = 0.9798.
05

Find Area to the Left of \(z=1.76\)

Using the Z-table, the area to the left of \(z=1.76\) is 0.9608.

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Most popular questions from this chapter

For the standard normal distribution, find the area within one standard deviation of the mean-that is, the area between \(\mu-\sigma\) and \(\mu+\sigma\).

During the 2009 edition of the reality show Britain's Got Talent, runner-up and Internet singing sensation Susan Boyle obtained \(20.2 \%\) of the first- place votes. Suppose that this percentage would hold true for all potential voters (note: the population of interest would be all viewers of ITV, which carries the show). Find the probability that, in a random sample of 250 potential voters, the number who would vote for Susan Boyle is a. exactly 57 b. 35 to 41 c. at least 60

Find the \(z\) value for each of the following \(x\) values for a normal distribution with \(\mu=30\) and \(\sigma=5\) a. \(x=39\) b. \(x=19\) c. \(x=24\) d. \(x=44\)

Determine the value of \(z\) so that the area under the standard normal curve a. in the right tail is 0250 b. in the left tail is \(.0500\) c. in the left tail is \(.0010\) d. in the right tail is. 0100

For the standard normal distribution, what is the area within three standard deviations of the mean?
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