Answer to Question #327645 in Statistics and Probability for Jennie

Question

A division-wide aptitude test in Mathematics was conducted to 1000 pupils. The mean of the test is 58 and the standard deviation is 12. The scores also approximate the normal distribution.

What is the minimum score to belong to the upper 10% of the group?

What are the two extreme scores outside of which 5% of the group and expected to fall?

What is the score that divides the distribution into two such that 75% of the cases is below it?

Estimate the range of scores that will include the:

Middle 50% of the distribution.

Middle 99% of the distribution.

Answer 1

Let the minimum score required to be in the upper 10% of the group be c. Then

$P(X \ge c)=0.10$

$P(X<c)=1-0.1=0.90$

$P(Z<\frac{c-58}{12})=0.9$

c-58=15.372

c=73.372

b. Let the minimum score required to be below 5% of the group be c.

$P(Z<\frac{c-58}{12})=0.05$

c-58=-19.74

c=38.26

Let the minimum score required to be in the upper 5% of the group be c.

$P(X \ge c)=0.05$

$P(X<c)=1-0.05=0.95$

$P(Z<\frac{c-58}{12})=0.95$

c-58=19.74

c=77.74

c. Let the minimum score required to be below 75% of the group be c.

$P(Z<\frac{c-58}{12})=0.75$

c-58=8.1

c=66.1

Middle 50% is contained in the range of 25%-75%

$P(Z<\frac{c-58}{12})=0.25$

c-58=-8.1

c=49.9

Then middle 50% is contained in the range of 49.9-66.1

Middle 50% is contained in the range of 0.5%-99.5%

$P(Z<\frac{c-58}{12})=0.005$

c-58=-30.9

c=27.1

$P(Z<\frac{c-58}{12})=0.995$

c-58=30.9

c=88.9

Middle 99% is contained in the range of 27.1% - 88.9%