The marks of the candidates in an entrance test are normally distributed with mean 65 and standard deviation 10. The candidates in the top 5% are given a scholarship. The minimum marks needed to get a scholarship are (Given \(P(0 \le Z \le 1.645) = 0.45\), where Z is the standard normal variate)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 81.45
Let x be the minimum marks. The top 5% lie above x, so
\(P(X \gt x) = 0.05\)
\(P(0 \le Z \le z) = 0.5 - 0.05 = 0.45\)
From the given value, z = 1.645. Using \(X = \mu + Z\sigma\) with \(\mu = 65\) and \(\sigma = 10\):
\(x = 65 + 1.645 \times 10\)
\(= 65 + 16.45 = 81.45\)
Why the other options are incorrect
Option (2) 48.55 — Incorrect. This is 65 − 16.45, the marks below which the lowest 5% lie.
Option (3) 69.5 — Incorrect. This is 65 + 0.45 × 10; the area 0.45 has been used as the value of z.
Option (4) 75 — Incorrect. This is one standard deviation above the mean (z = 1); the cut-off for 5% needs z = 1.645.