Question

Assume that scores on a math competition are normally distributed a mean of 12 points and a standard deviation of 2.5 points. Answer the following questions on these scores

How many students scored less than 9 points? Round to the nearest percent.

How many students scored at least 17 points? Round to the nearest percent.

How many students have a score between 12 and 15 points? Round to the nearest percent.

What is the cuttoff score for top 5% students. Round to the nearest whole number.

Choose a sample of 25 scores randomly, what is the probability the sample mean is less then 11.5 points? Round to two decimal places.

Answer 1

(a)

= 12

= 2.5

To find P(X<9):

Z = (9 - 12)/2.5 = - 1.20

Table of Area Under Standard Normal Curve gives area = 0.3849

So,

P(X<9) = 0.5 - 0.3849 = 0.1151 = 12 %

So,

Answer is:

12 %

(b)

= 12

= 2.5

To find P(X17):

Z = (17 - 12)/2.5 = 2

Table of Area Under Standard Normal Curve gives area = 0.4772

So,

P(X<9) = 0.5 - 0.4772 = 0.0228 = 2 %

So,

Answer is:

2 %

(c)

= 12

= 2.5

To find P(12<X<15):

Z = (15 - 12)/2.5 = 1.20

Table of Area Under Standard Normal Curve gives area = 0.3849

So,

P(12<X<15) = 0.3849 = 38 %

So,

Answer is:

38 %

(d)'

Top 5% corresponds to area = 050 - 0.05= 0.45 from mid value to Z on RHS.

Table gives Z = 1.645

So,

Z = 1.645 = (X - 12)/2.5

So

X = 12 + (1.645 X 2.5) = 16.1125 = 16, round to nearest whole number.

So,

Answer is:

16

(e)

= 12

= 2.5

n = 25

SE = /

= 2.5/ = 0.5

To find P(X<11.5):

Z = (11.5- 12)/0.5 = - 1

Table of Area Under Standard Normal Curve gives area = 0.3413

So,

P(X<9) = 0.5 - 0.3413 = 0.1587

So,

Answer is:

0.16