Question

Q2. A college that specialises in training people for top-level management positions administers various types of physical and psychological tests. In one such test, objects must be assembled and pictures arranged in order. The college has found that results in the past have followed a normal distribution with a mean of 91.7 and a standard deviation of 8.3. a. What percentage of people obtain scores in the following ranges:

i) Above 105

ii) Between 100 and 110

b. Suppose that only 5% of all people obtain better scores that Bill. What is Bill’s score?

Answer 1

Solution:

Let X be a random variable which represents the scores of test.

Given that, X ~ N(91.7, 8.3²)

i.e.  μ = 91.7 and σ = 8.3

a(i) We have to obtain P(X > 105)

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

Using "pnorm" function of R we get, P(Z > 1.6024) = 0.0545

0.0545 = 5.45%

5.45% of people obtain scores above 105.

a(ii) We have to obtain P(100 < X < 110).

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

Using "pnorm" function of R we get,

P(Z > 2.2048) = 0.9863 and P(Z < 1) = 0.8413

0.14550 = 14.50%

14.50% of people obtain scores between 100 and 110.

b) Let Bill's score is k.

Hence, P(X > k) = 0.05

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

............................(1)

Using "qnorm" function of R we get P(Z > 1.64485) = 0.05

Comparing, P(Z > 1.64485) = 0.05 and (1) we get,

Hence, Bill's score is 105.3523.

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