Question

Assume that the average score of the Final exam of a particular class is μ = 82.0 and standard deviation σ = 15. The probability (proportion) of a student selected at random to have a score more than 90 on the exam is approximately:
A) 0.296
B) 0.704
C) 0.180
D) 0.819
E) None of the above

Answer 1

To find the probability that a randomly selected student scores more than 90 on the exam, we can use the properties of the normal distribution.

Given:
- Mean (μ) = 82.0
- Standard deviation (σ) = 15.0
- Score (X) = 90

We can standardize the score to find the corresponding z-score using the formula:

\[
z = \frac{X - \mu}{\sigma}
\]

Substituting the values:

\[
z = \frac{90 - 82}{15} = \frac{8}{15} \approx 0.5333
\]

Next, we will look up the z-score of approximately 0.53 in the standard normal distribution table, or we can use a calculator. The z-table tells us the probability that a value is less than a certain z-score.

For \( z = 0.53 \), the cumulative probability (P(Z < 0.53)) is approximately 0.7027.

To find the probability of a score greater than 90, we calculate:

\[
P(X > 90) = 1 - P(Z < 0.53) = 1 - 0.7027 \approx 0.2973
\]

Rounding this result, we find:

\[
P(X > 90) \approx 0.296
\]

Therefore, the correct answer is:

A) 0.296