Question

Suppose that from past data a professor knows that the test score of a typical student taking their final examination is a normal random variable with mean 73 and standard deviation 10. (a) If 5 students are selected at random, what is the probability that their sample average grade will be within 3 of 73? (b) What is the minimum number of students that need to take the exami- nation to ensure, with probability at least 0.95, that the class average would be within 3 of 73?

Answer 1

a)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	73
std deviation   =σ=	10.0000
sample size       =n=	5
std error=σx̅=σ/√n=	4.4721

probability that their sample average grade will be within 3 of 73 :

probability =

P(70<X<76)

=

P(-0.67<Z<0.67)=

0.7486-0.2514=

0.4972

b)

for 95 % CI value of z=	1.960
standard deviation σ=	10.00
margin of error E =	3
required sample size n=(zσ/E)2                                         =	43.0