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Question
Entry to a certain University is determined by a national test. The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100. Raghul wants to be admitted to this university and he knows that he must score better than at least 70% of the students who took the test. Raghul takes the test and scores 585. Will he be admitted to this university?
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Solution
Let x denotes the scores of a national test mean
µ = 500 and standard deviation σ = 100
Standard normal variate z = `(x - mu)/sigma = (x - 5000)/100`
When x = 585
z = `(585 - 500)/100 = 85/100` = 0.85
P(X ≤ 585) = P(Z ≤ 0.85)
P(Z ≤ 0.85) = P(`-oo` < z < 0) + P(0 < z < 0.85)
= 0.5 + 0.3023
= 0.8023
For n = 100
P(Z ≤ 0.85) = 100 × 0.8023
= 80.23
∴ Raehul scores 80.23%
We can determine the scores of 70% of the students as follows:
From the table for the area 0.35
We get z1 = – 1.4(as z1 lies to left of z = 0)
Similarly z2 = 1.4
Now z1 = `(x_1 - 500)/100`
⇒ – 1.4 = `(x_1 - 500)/100`
– 1.4 × 100 = x1 – 500
⇒ x1 500 – 140
x1 = 360
Again z2 = `(x_2 - 500)/100`
⇒ – 1.4 = `(x_1 - 500)/100`
1.4 × 100 = x2 – 500
⇒ x2 = 140 + 500
= x2 = 640
Hence 70% of students score between 360 and 640
But Raghul scored 585. His score is not better than the score of 70% of the students.
∴ He will not be admitted to the university.
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