Sure! Let's solve this step-by-step.
### Step-by-Step Solution
1. Understand the Given Values:
- The mean score [tex]\( (\mu) \)[/tex] is 165.
- The standard deviation [tex]\( (\sigma) \)[/tex] is 13.
- The total number of games [tex]\( (N) \)[/tex] is 90.
- The target score is 193.
2. Calculate the Z-Score:
- The Z-score formula is:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
- Plug in the values [tex]\( X = 193 \)[/tex], [tex]\( \mu = 165 \)[/tex], and [tex]\( \sigma = 13 \)[/tex]:
[tex]\[
Z = \frac{193 - 165}{13} = \frac{28}{13} \approx 2.15
\][/tex]
3. Find the Probability:
- We will use the cumulative distribution function (CDF) of the normal distribution to find the probability that a score is less than the Z-score calculated.
- The CDF value corresponding to a Z-score of approximately 2.15 is 0.9844 (rounded to four decimal places).
4. Calculate the Expected Number of Games:
- To find the expected number of games with a score less than 193, multiply the total number of games [tex]\( (N) \)[/tex] by the probability obtained:
[tex]\[
\text{Expected number of games} = N \times \text{Probability} = 90 \times 0.9844 \approx 88.59
\][/tex]
5. Round to the Nearest Whole Number:
- 88.59 rounded to the nearest whole number is 89.
### Conclusion
Out of the 90 games that Nevaeh bowled last year, she is expected to score less than 193 in approximately 89 of them.
