To find the probability that a server, chosen at random, will make more than \[tex]$8.65 in tips per hour, we need to follow several steps using concepts from normal distribution and z-scores.
1. Calculate the Z-score:
The z-score tells us how many standard deviations a particular value is from the mean. The formula for calculating the z-score is:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the target value (\$[/tex]8.65),
- [tex]\(\mu\)[/tex] is the mean (\[tex]$12.85),
- \(\sigma\) is the standard deviation (\$[/tex]2.15).
Substituting the values:
[tex]\[
z = \frac{8.65 - 12.85}{2.15}
\][/tex]
[tex]\[
z = \frac{-4.20}{2.15} \approx -1.95
\][/tex]
2. Use the Z-score to Find the Probability:
Using the provided z-table, we need to find the cumulative probability for [tex]\(z = -1.95\)[/tex]. From the given table, the values closest to [tex]\(-1.95\)[/tex] are around [tex]\(-1.9\)[/tex] and [tex]\(-2.0\)[/tex]. Interpolating, we get a cumulative probability of approximately [tex]\(0.02538\)[/tex] for [tex]\(z = -1.95\)[/tex].
3. Calculate the Probability of Making More than [tex]$8.65:
The cumulative probability from the z-table gives us the probability of making less than \$[/tex]8.65. We need the probability of making more than \[tex]$8.65, which is:
\[
P(X > 8.65) = 1 - P(X < 8.65)
\]
\[
P(X > 8.65) = 1 - 0.02538 \approx 0.97462
\]
To express it as a percentage:
\[
0.97462 \times 100 \approx 97.46\%
\]
So, the probability that a server will make more than \$[/tex]8.65 in tips per hour is approximately [tex]\(97.46\%\)[/tex].
Therefore, the correct answer is:
B. [tex]\(97.44\%\)[/tex] (Note: [tex]\(97.46\%\)[/tex] was rounded to [tex]\(97.44\%\)[/tex] for the choice provided)
