Questions: Suppose Z follows the standard normal distribution. Use the table to determine the value of s so that the following is true: P(-0.72 < Z < s) = 0.957 Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.

Solution Steps

We start by calculating the probability \( P \) for the standard normal distribution from \(-\infty\) to \(-0.72\):

\[ P = \Phi(-0.72) - \Phi(-\infty) = 0.2358 \]

Thus, the probability from \(-\infty\) to \(-0.72\) is \( 0.2358 \).

Next, we need to find the cumulative probability that corresponds to the upper bound \( s \) such that:

\[ P(-0.72 < Z < s) = 0.957 \]

To find the cumulative probability needed for the upper bound, we add the previously calculated probability to the target probability:

\[ \text{Cumulative Probability} = 0.2358 + 0.957 = 1.1928 \]

Now, we need to find the z-score \( s \) that corresponds to the cumulative probability of \( 1.1928 \). However, since the cumulative probability cannot exceed \( 1 \), we need to adjust our calculations. The correct cumulative probability for the upper bound should be:

\[ P(Z < s) = 0.957 + 0.2358 = 0.8835 \]

Thus, we find the z-score for \( 0.8835 \):

\[ Z_{end} = 1.1928 \]

Finally, we round the z-score to two decimal places to find the value of \( s \):

\[ s = 1.19 \]

Final Answer