Question

Use normal approximation (without continuity correction) to estimate the probability of passing a true/false test of 30 questions if the minimum passing grade is 60% and all responses are random guesses.

Answer 1

Here' how to solve the question. please go step-by-step. Write back in case you've doubts.

Lets first define the event as : X be the random number of correct answers out of n = 30 true/false questions.

The probability a single question is answered correctly is p=0.5. Thus the exact distribution of X is binomial:

X∼Binomial(n=30,p=0.5).

In order to get a passing grade of at least 60%, the required number of correct responses is X≥(0.6)(30)=18.

we will convert binomial distribution to normal distribution ( approximated)

X∼∙Normal(μ=n*p,σ=sqrt(n*p*(1−p))) [We wll use the Z = (X-mean)/Standard deviation to calculate Z score]

So,
Pr(X>= 18) ~ Pr[X>= (18-n*p)/sqrt(n*p*(1-p))]
= Pr[Z >= (18-30*.5)/sqrt(30*.5*.5)]
= Pr[Z> 1.095]
= 0.1367

Answer: 0.1367