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A national restaurant chain claims that their servers make an average of [tex]\$12.85[/tex] in tips per hour, with a standard deviation of [tex]\$2.15[/tex]. Given that the data is approximately normal, find the probability that a server, chosen at random, will make more than [tex]\[tex]$8.65[/tex] in tips per hour.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{8}{|c|}{ Table shows values to the LEFT of the $[/tex]z[tex]$-score } \\
\hline
$[/tex]z$ & 0.02 & 0.03 & 0.04 & 0.05 & 0.06 & 0.07 & 0.08 & 0.09 \\
\hline
1.6 & 0.94738 & 0.94845 & 0.94950 & 0.95053 & 0.95154 & 0.95254 & 0.95352 & 0.95449 \\
\hline
1.7 & 0.95728 & 0.95818 & 0.95907 & 0.95994 & 0.96080 & 0.96164 & 0.96246 & 0.96327 \\
\hline
1.8 & 0.96562 & 0.96638 & 0.96712 & 0.96784 & 0.96856 & 0.96926 & 0.96995 & 0.97062 \\
\hline
1.9 & 0.97257 & 0.97320 & 0.97381 & 0.97441 & 0.97500 & 0.97558 & 0.97615 & 0.97670 \\
\hline
2.0 & 0.97831 & 0.97882 & 0.97932 & 0.97982 & 0.98030 & 0.98077 & 0.98124 & 0.98169 \\
\hline
-2.0 & 0.02169 & 0.02118 & 0.02068 & 0.02018 & 0.01970 & 0.01923 & 0.01876 & 0.01831 \\
\hline
-1.9 & 0.02743 & 0.02680 & 0.02619 & 0.02559 & 0.02500 & 0.02442 & 0.02385 & 0.02330 \\
\hline
-1.8 & 0.03438 & 0.03362 & 0.03288 & 0.03216 & 0.03144 & 0.03074 & 0.03005 & 0.02938 \\
\hline
-1.7 & 0.04272 & 0.04182 & 0.04093 & 0.04006 & 0.03920 & 0.03836 & 0.03754 & 0.03673 \\
\hline
-1.6 & 0.05262 & 0.05155 & 0.05050 & 0.04947 & 0.04846 & 0.04746 & 0.04648 & 0.04551 \\
\hline
\end{tabular}

A. [tex]10.75 \%[/tex]
B. [tex]97.44 \%[/tex]
C. [tex]2.56 \%[/tex]
D. [tex]89.25 \%[/tex]


Sagot :

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)
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