Question

From past experience an instructor knows that the score of a
student taking his final examination is a random variable with mean
53.6 and standard deviation 18.5. Assume that the score is normally
distributed. (a) What is the probability that a student can get a
score larger than 60? (b) What is the probability that the average
score of the students in a class of size 72 exceeds 60? (c) What
passing mark should he set such that 95% of the students in his
class will pass the examination? (d) If the normal assumption is
violated (i.e. the actual distribution of the score is not normal),
will the calculations in (a), (b) and (c) still be valid? Why?

Answer 1

Let X denote the score of a student taking his final examination. Assuming X is normally distributed,

Standardizing the random variable X,

  

(a) To find : The probability that a student can get a score larger than 60

From Standard normal table, the required probability can be obtained by looking for the area corresponding to the Z score 0.35 (Since, standard normal table gives the area to the left of the Z score)

= 1 - 0.63683

= 0.36317

Hence, the probability that a student can get a score larger than 60.

(b) To find: The probability that the average score of the students in a class of size 72 exceeds 60.

By central limit theorem, for sufficiently large sample size (usually > 30), the sample mean, say of a normally distributed random variable X with mean and standard deviation is also approximately normally distributed with mean and standard deviation ( n = Sample size)

For n = 72,

................(Since, )

From standard normal table,

= 1 - 0.99836

= 0.0016

(c) By definition of percentile, i.e. the percentage of values that lie below it, the passing mark such that 95% of the students in the class will pass the examination is nothing but the 5th percentile, i.e only 5% of the students fall below the passing mark.

From standard normal table, the 5th percentile gives the Z score Z = -1.645

Hence, the passing mark should he set as 23, such that 95% of the students in his class will pass the examination.

(d) If the actual distribution of X is not normal, but if the sample size is 72 > 30 (As mentioned in (b)),

By central limit theorem, for sufficiently large random sample (usually > 30), the sampling distribution of the sample mean of the random variable is approximately normally distributed, irrespective of the actual distribution of the variable.

Hence, we may say that the calculations in (b) would still be valid if the normality assumption is violated, for a large sample (n = 72).