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Chapter 6: Problem 21

Determine the following probabilities for the standard normal distribution. a. \(P(-1.83 \leq z \leq 2.57)\) b. \(P(0 \leq z \leq 2.02)\) c. \(P(-1.99 \leq z \leq 0)\) d. \(P(z \geq 1.48)\)

Short Answer

Expert verified
The probabilities are: for a, the area between -1.83 and 2.57; for b, the area between 0 and 2.02; for c, the area between -1.99 and 0; and for d, the area bigger than 1.48 in a standard normal distribution. The exact values would depend on the specific standard normal table used.

Step by step solution

01

Find the probabilities for each z-value

Use the standard normal table to find the area to the left of the given z-values. For a, look up the values for -1.83 and 2.57. For b, look up the values for 0 and 2.02. For c, look up the values for -1.99 and 0. For d, look up the value for 1.48.
02

Determine the probabilities

For a, b, and c, the probability is determined by subtracting the smaller z-value from the larger z-value. For d, the probability is found by subtracting the area from 1 since the question asks for the probability when z is greater than 1.48, and the standard normal table always gives the area to the left (i.e., probabilities for Z <= z).
03

Interpret the Results

Each of these probabilities represents the likelihood of the random variable Z falling within the described range in a standard normal distribution. The closer the probability is to 1, the higher the likelihood, the closer it is to 0, the lower the likelihood.

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