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Chapter 6: Problem 88

Find the following probabilities for the standard normal random variable \(z\) : a. \(P(-1.96 \leq z \leq 1.96)\) b. \(P(z>1.96)\) c. \(P(z<-1.96)\)

Short Answer

Expert verified
Answer: The probabilities are as follows: - P(-1.96 鈮� z 鈮� 1.96) = 0.9500 - P(z>1.96) = 0.0250 - P(z<-1.96) = 0.0250

Step by step solution

01

Identify area under curve between -1.96 and 1.96

In this case, we want to find the area under the curve between \(z = -1.96\) and \(z= 1.96\). To do this, we can use the standard normal distribution table to locate the probability to the left of each of these values, and then subtract the two probabilities.
02

Locate probabilities in z-table

First, locate the probability to the left of \(z = 1.96\). According to the z-table, this value is \(0.9750\). Then, locate the probability to the left of \(z = -1.96\). This value is \(0.0250\).
03

Subtract probabilities

Now, subtract the probability for \(z = -1.96\) from the probability for \(z = 1.96\): \(P(-1.96 \leq z \leq 1.96) = 0.9750 - 0.0250 = 0.9500\) #b. Finding P(z>1.96)#
04

Identify area under curve to the right of 1.96

In this case, we want to find the area under the curve to the right of \(z = 1.96\). As the standard normal distribution table provides probabilities to the left of \(z\) values, we need to find the complement of the probability to the left of the \(z = 1.96\) value.
05

Subtract probability from 1

The probability to the left of \(z = 1.96\) is \(0.9750\). To find the area to the right of \(z = 1.96\), subtract this value from \(1\): \(P(z>1.96) = 1 - 0.9750 = 0.0250\) #c. Finding P(z<-1.96)#
06

Locate probability in z-table

In this case, we want to find the area under the curve to the left of \(z = -1.96\). As our z-table provides probabilities for this directly, we can simply use the value from the z-table.
07

Write down result

The probability to the left of \(z = -1.96\) is \(0.0250\). Therefore, the probability is: \(P(z <-1.96) = 0.0250\).

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