Questions: Using the standard normal distribution, find the following probabilities: [ P(-0.58<z<2.43) ]

Solution Steps

To find the probability \( P(-0.58 < z < 2.43) \) using the standard normal distribution, we can express it in terms of the cumulative distribution function \( \Phi \): \[ P(-0.58 < z < 2.43) = \Phi(2.43) - \Phi(-0.58) \]

The Z-scores for the bounds are:

• For the lower bound: \( z_{start} = -0.58 \)
• For the upper bound: \( z_{end} = 2.43 \)

Using the cumulative distribution function values: \[ \Phi(2.43) \approx 0.9236 \quad \text{and} \quad \Phi(-0.58) \approx 0.2810 \]

Now, substituting the values into the probability expression: \[ P(-0.58 < z < 2.43) = 0.9236 - 0.2810 = 0.6426 \]

Rounding the final probability to four decimal places gives: \[ P(-0.58 < z < 2.43) \approx 0.7115 \]

Final Answer