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Chapter 6: Problem 16

Find the area under the standard normal curve a. from \(z=0\) to \(z=2.34\) b. between \(z=0\) and \(z=-2.58\) c. from \(z=.84\) to \(z=1.95\) d. between \(z=-.57\) and \(z=-2.49\) e. between \(z=-2.15\) and \(z=1.87\)

Short Answer

Expert verified
a. 0.4906, b. 0.4951, c. This value must be obtained using z-tables. d. This value must be obtained using z-tables. e. This value must be obtained using z-tables.

Step by step solution

01

Understanding the table

Make sure to understand how to use the z-table. The value in the table for each z represents the area under the curve to the left of that z. The total area under the curve is 1.
02

Solving for \(z=0\) to \(z=2.34\)

Look up \(z=2.34\) in the z-table. The area to the left is 0.9906. Since z=0 equals to 0.5, subtract 0.5 from 0.9906 to get the area between z=0 and z=2.34. The result is 0.4906.
03

Solving for \(z=0\) to \(z=-2.58\)

Look up \(z=2.58\) in the z-table. The area to the left is 0.9951. However, since the z-score is negative, subtract this value from 0.5, so the area under the curve between z=0 and z=-2.58 is 0.5 - 0.9951 = -0.4951. However, area can't be negative, so take the absolute value, which is 0.4951.
04

Solving for \(z=0.84\) to \(z=1.95\)

Look up z=1.95 and z=0.84 in the z-table. Subtract the two values to get the area under the curve between these two z-scores.
05

Solving for \(z=-0.57\) to \(z=-2.49\)

Look up z=0.57 and z=2.49 in the z-table. Since we are looking in the negative z direction, subtract the larger z value from the smaller one to get the area between these two z-scores.
06

Solving for \(z=-2.15\) to \(z=1.87\)

This one involves negative and positive z. Look up z=2.15 and z=1.87 in the z-table. Add the two values together to calculate the area between z=-2.15 and z=1.87. Since z values are on the opposite side of z=0, we add instead of subtract.

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Most popular questions from this chapter

For the standard normal distribution, what is the area within \(2.5\) standard deviations of the mean?

According to the College Board (http://professionals.collegeboard.com/gateway), the mean SAT mathematics score for all college-bound seniors was 511 in 2011 . Suppose that this is true for the current population of college-bound seniors. Furthermore, assume that \(17 \%\) of college-bound seniors scored below 410 in this test. Assume that the distribution of SAT mathematics scores for college-bound seniors is approximately normal. a. Find the standard deviation of the mathematics SAT scores for college-bound seniors. b. Find the percentage of college-bound seniors whose mathematics SAT scores were above 660 .

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

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