To determine how many of the 100 people would give the movie a score above 7.3 points, we can follow these steps:
1. Identify the problem parameters:
- Population mean ([tex]\(\mu\)[/tex]): 8.3
- Population standard deviation ([tex]\(\sigma\)[/tex]): 0.5
- Sample size: 100
- Threshold score: 7.3
2. Calculate the Z-score for the threshold score:
The Z-score formula is given by:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\( X \)[/tex] is the threshold score.
Plugging in our values:
[tex]\[
Z = \frac{7.3 - 8.3}{0.5} = -2.0
\][/tex]
3. Determine the cumulative probability for the Z-score:
Using the Z-score, we can determine the cumulative probability up to that score, which tells us the proportion of the population scoring below 7.3. The cumulative probability for a Z-score of -2.0 is approximately 0.02275.
4. Calculate the probability above the threshold:
To find the probability of someone scoring above 7.3, we subtract the cumulative probability from 1:
[tex]\[
\text{Probability above 7.3} = 1 - 0.02275 = 0.97725
\][/tex]
5. Calculate the expected number of people scoring above 7.3:
Multiply this probability by the sample size:
[tex]\[
\text{Expected number of people} = 0.97725 \times 100 = 97.725
\][/tex]
6. Round to the nearest whole person:
Rounding 97.725 gives us 97.
Therefore, we expect that approximately 97 out of the 100 people would give the movie a score above 7.3 points.
