Question

College entrance examination scores (X) have an approximately normal distribution with mean 500 and standard deviation 100.

(b) If 4 scores are selected at random, what is the distribution of their average? Include the name of the distribution and the values of any parameters. Show work if possible.

(c) If 4 scores are selected at random, what is the probability that their average is 510 or greater?

(d) If 25 scores are selected at random, what is the probability that their average is 510 or greater?

(e) What would be the distribution of the difference between the sample averages between parts (c) and (d)? Use the difference `average in (c) - average in (d).` Include the name of the distribution and the values of any parameters.

Answer 1

X : Entrance examination scores

X follows a normal distribution with mean = 500 and standard deviation =100

By central limit theorem,

Sample mean (for a sample size of n) follows a normal distribution with mean =500 and standard deviation :

(b)

If 4 scores are selected at random, i.e Sample size n=4;

distribution of their average : follows normal distribution with mean = 500 and standard deviation = 100/ = 100/2 =50

(c)

If 4 scores are selected at random, probability that their average is 510 or greater = P( > 510)

P( > 510) = 1 - P(510)

Z-score for 510 = (510-500)/50 = 0.2

From standard normal tables, P(Z0.2) = 0.5793

P(510) = P(Z0.2) = 0.5793

P( > 510) = 1 - P(510) = 1 -0.5793=0.4207

If 4 scores are selected at random, probability that their average is 510 or greater = 0.4207

(d) If 25 scores are selected at random, probability that their average is 510 or greater

If 25 scores are selected at random, i.e Sample size n=25

distribution of their average : follows normal distribution with mean = 500 and standard deviation = 100/ = 100/5 =20

probability that their average is 510 or greater = P( > 510)

Z-score for 510 = (510-500)/20 = 10/20 = 0.5

From standard normal tables, P(Z0.5) = 0.6915

P(510) = P(Z0.2) = 0.6915

P( > 510) = 1 - P(510) = 1 -0.6915=0.3085

If 25 scores are selected at random, probability that their average is 510 or greater = 0.3085

(e)

What would be the distribution of the difference between the sample averages between parts (c) and (d)? Use the difference `average in (c) - average in (d).` Include the name of the distribution and the values of any parameters.

d : Difference between the sample averages between parts (c) and (d)(average in (c) - average in (d))

i.e

d = -

follows normal distribution with mean = 500 and standard deviation = 50

follows normal distribution with mean = 500 and standard deviation = 20

d( - ) also follows normal distribution with mean = - and standard deviation

= - = 500-500=0

d( - ) also follows normal distribution with mean =0 and standard deviation =53.85164807

i.e

the difference `average in (c) - average in (d).`  normal distribution with mean =0 and standard deviation =53.85164807