Question

© macmillan learning use a standard normal table to determine the probability. give your answer as a decimal to four decimal places. \\( p(-1.5 < z < 1.5) = \\)

Explanation:

Step1: Recall Z - table usage

To find \( P(-1.5 < z < 1.5) \), we use the property of the standard normal distribution. The probability that \( z \) is between \( -a \) and \( a \) is \( P(-a < z < a)=2P(0 < z < a)- 1+2P(z < 0) \), but more simply, since \( P(z < a)-P(z < -a) \), and for standard normal \( P(z < -a)=1 - P(z < a) \). So \( P(-1.5 < z < 1.5)=P(z < 1.5)-P(z < -1.5) \). And \( P(z < -1.5)=1 - P(z < 1.5) \), so \( P(-1.5 < z < 1.5)=2P(z < 1.5)-1 \).

Step2: Find \( P(z < 1.5) \) from Z - table

Looking up \( z = 1.5 \) in the standard normal table, we find that \( P(z < 1.5)=0.9332 \).

Step3: Calculate the probability

Substitute \( P(z < 1.5) = 0.9332 \) into the formula: \( P(-1.5 < z < 1.5)=2\times0.9332 - 1 \).
First, calculate \( 2\times0.9332=1.8664 \).
Then, \( 1.8664 - 1 = 0.8664 \).

Answer:

\( 0.8664 \)