Question

Question 21 (2.34375 points) The mean composite ACT score for graduating seniors between 2013 and 2015 was 21.0 with a standard deviation of 5.4. Assume the composite ACT scores are normally distributed. To qualify for a scholarship at the University of Georgia, your ACT score must be in the top 7%. Find the needed ACT score. Round to a whole number A/

Answer 1

Solution :

Let X represents the composite ACT scores.

Given that, X ~ N(21.0, 5.4²)

μ = 21.0 and  σ = 5.4

To qualify for the University of Georgia, one's ACT score must be in top 7%. Hence, the minimum ACT score will be the corresponding to 93th percentile.

Let the minimum ACT score needed is k.

Hence, P(X < k) = 0.93

We know that if X ~ N(μ, σ²) then,

X-M Z=1 ~ N(0,1) o

X- k-u ::P 0.93 o

.:PZ < P(2 k – 21.0 5.4 = = 0.93 ............(1)

Using "qnorm" function of R we get, P(Z < 1.476) = 0.93

Comparing, P(Z < 1.476) = 0.93 and (1) we get,

h-21.0 5.4 - 1.476

k= 21.0 + (1.476 x 5.4)

< 29

Hence, the needed ACT score is 29.

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