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Chapter 6: Problem 22

Determine the following probabilities for the standard normal distribution. a. \(P(-2.46 \leq z \leq 1.88)\) b. \(P(0 \leq z \leq 1.96)\) c. \(P(-2.58 \leq z \leq 0)\) d. \(P(z \geq .73)\)

Short Answer

Expert verified
a. The probability is approximately 0.0231.\nb. The probability is approximately 0.475.\nc. The probability is approximately 0.495.\nd. The probability is approximately 0.2327.

Step by step solution

01

Identify Probabilities

Recognize that the areas under the normal curve to the left or right of a particular z score, or between two z scores, correspond to probabilities. Use standard normal distribution (z-distribution) tables and formulas to find the probabilities.
02

Calculate Probability for Part (a)

The probability \(P(-2.46 \leq z \leq 1.88)\) is the area under the curve to the right of z = -2.46 and to the left of z = 1.88. Refer to the standard normal distribution table. The entry for z = 2.46 (ignoring the sign) is 0.9930, and the entry for z = 1.88 is 0.9699. Subtract these to get the probability for the range -2.46 <= z <= 1.88: \(0.9930 - 0.9699 = 0.0231\).
03

Calculate Probability for Part (b)

For \(P(0 \leq z \leq 1.96)\), this is the area under the curve to the right of z = 0 (which is where the curve peaks and therefore includes half the total area, or 0.5) and to the left of z = 1.96. From the table, the entry for z = 1.96 is 0.975. So subtract 0.5 from this to find the probability for the range 0 <= z <= 1.96: \(0.975 - 0.5 = 0.475\).
04

Calculate Probability for Part (c)

The probability \(P(-2.58 \leq z \leq 0)\) requires finding the area under the curve to the right of z = -2.58 and to the left of z = 0. From the table, the entry for z = 2.58 (again ignoring the sign, as the distribution is symmetrical) is 0.995. So subtract 0.5 (for the area to the left of z = 0) from this to find the probability for the range -2.58 <= z <= 0: \(0.995 - 0.5 = 0.495\).
05

Calculate Probability for Part (d)

The probability \(P(z \geq .73)\) requires finding the area under the curve to the right of z = 0.73. From the table, the entry for z = 0.73 is 0.7673. However, the table gives you the area to the left of the given z-score, so to get the area to the right of z = 0.73, subtract the table value from 1: \(1 - 0.7673 = 0.2327\).

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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?

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