Questions: If z is a standard normal variable, find the probability that z lies between -2.41 and 0. Round to four decimal places. A. 0.5080 B. 0.4920 C. 0.0948 D. 0.4910

Solution Steps

We need to find the probability that a standard normal variable \( z \) lies between \(-2.41\) and \(0\). This can be expressed mathematically as:

\[ P(-2.41 < z < 0) = \Phi(0) - \Phi(-2.41) \]

where \( \Phi(z) \) is the cumulative distribution function (CDF) of the standard normal distribution.

The Z-scores for the bounds are:

• For the lower bound: \[ Z_{start} = -2.41 \]
• For the upper bound: \[ Z_{end} = 0.0 \]

Using the properties of the standard normal distribution:

• \( \Phi(0) = 0.5 \)
• \( \Phi(-2.41) \) can be found using standard normal distribution tables or calculators, which gives approximately \( 0.0080 \).

Now we can compute the probability:

\[ P(-2.41 < z < 0) = \Phi(0) - \Phi(-2.41) = 0.5 - 0.0080 = 0.492 \]

Final Answer