Questions: Listen If z is a standard normal variable, find the probability. The probability that z lies between 0.7 and 1.98 is .

Listen If z is a standard normal variable, find the probability. The probability that z lies between 0.7 and 1.98 is .
Transcript text: Listen If $z$ is a standard normal variable, find the probability. The probability that $z$ lies between 0.7 and 1.98 is $\qquad$ .
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Solution

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Solution Steps

To find the probability that a standard normal variable \( z \) lies between 0.7 and 1.98, we need to calculate the cumulative distribution function (CDF) values for both 0.7 and 1.98 and then find the difference between these two values. This will give us the probability that \( z \) falls within this range.

Step 1: Calculate the Cumulative Distribution Function (CDF) for \( z = 0.7 \)

The cumulative distribution function for a standard normal variable at \( z = 0.7 \) is calculated as: \[ P(Z \leq 0.7) = 0.7580 \]

Step 2: Calculate the CDF for \( z = 1.98 \)

The cumulative distribution function for a standard normal variable at \( z = 1.98 \) is calculated as: \[ P(Z \leq 1.98) = 0.9761 \]

Step 3: Calculate the Probability that \( z \) is Between 0.7 and 1.98

The probability that the standard normal variable \( z \) lies between 0.7 and 1.98 is the difference between the two CDF values: \[ P(0.7 < Z < 1.98) = P(Z \leq 1.98) - P(Z \leq 0.7) = 0.9761 - 0.7580 = 0.2181 \]

Final Answer

\(\boxed{0.2181}\)

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