Question

For the large population of students who attend Zarthan University, the distribution of the variable Math ACT Score is normal with a mean of 25.8 and a standard deviation of 2.6.  

a. What proportion of students obtain a Math ACT Score of 28 or higher? (5 decimals)

b. If 3 students are selected at random, what is the probability that all 3 have a Math ACT Score of 28 or higher? (5 decimals)

c. Compared with the probability correctly computed in Part 2, the probability that the mean Math ACT Score of the 3 students is 28 or higher would be:

A. the same B. smaller C. larger D. insufficient info given

d. What is the 99th percentile of the Math ACT Score distribution? (2 decimals)

e. At Zartman University the distribution of Math ACT scores has a higher mean and higher standard deviation than at Zarthan University. Compared with the proportion of students at Zarthan University having a Math ACT score above 28, the proportion at Zartman is:

     A. the same B. smaller C. larger D. insufficient info given

f. At Burnshortz University the distribution of Math ACT scores has a higher mean but lower standard deviation than at Zarthan University. Compared with the proportion of students at Zarthan University having a Math ACT score above 28, the proportion at Burnshortz is:

     A. the same B. smaller C. larger D. insufficient info given

Answer 1

A.

P(X>28) = P(Z>(28-25.8)/2.6)

= P(Z>0.846)

= 0.1987

B.

Probability = 0.1987³

= 0.00785

C.

P(>28) = P(Z>(28-25.8)/2.6/)

= P(Z>1.466)

= 0.0712

i.e. C. larger

D.

P(Z<z) = 0.99

then

z = 2.3263

So,

99th percentile score = 25.8 + 2.6*2.3263

= 31.8484

E.

We know that,

Z = (X-mean)/std. deviation

So, if mean and std. deviation increases and X remains the same then:

Z will decrease and so,

Compared with the proportion of students at Zarthan University having a Math ACT score above 28, the proportion at Zartman is:

C. larger

F.

D. insufficient info given

As we can't know, which way the z-value will go, up or down.

Please upvote if you have liked my answer, would be of great help. Thank you.

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