Question

On a math test, a student answers 2 problems on average in 40 minutes a) What is the probability that the number of problems answered in 60 minutes is more than 2? b) What is the expected value and the standard deviation of the number of problems answered in 70 minutes?

Answer 1

Given the number of problems answered = 2 in 40 minutes,

The most appropriate model for this is poisson distribution, so number of problems answered follows a poisson process.

So average number of problems answered in 60 min = 2*40/60 = 8/6 = 8/6

So let X be the number of problems aswered in 60 min then X ~ Poisson(8/6)

Need to find P(X > 2)

Now P(X > 2) = 1 - P(X = 0) - P(X = 1) - P(X = 2)

Now for poisson distribution

Calculating this for x = 0,1,2 and substituting in the expression

P(X > 2) = 0.15065

b)

Since 2 problems answered on average in 40 minutes

So number of problem in 70 minutes = 2*70/60 = 14/6 = 7/3

Hence if Y =. number of problem answered in 70 minutes then Y ~ Poisson(7/3)

and hence E(Y) = 7/3

Var(Y) = 7/3

So SD(Y) = sqrt(7/3) = 1.527