Question

A class has a total number of 46 students. The average number of absences per lecture is 13, with a standard deviation of 7 absences. Show that there is at most a 25% chance that the number of attendances next lecture is less than 20.

[Hint: You may consider Markov's inequality or Chebyshev's inequality.]

Answer 1

Let X be the number of absences then

E(X) = 13

SD(X) = 7

Need to show P(46 - X < 20) < 0.4 or P(X > 26) < 0.4

Now P(X > 26) = P(X - 13 > 13) <= P( |X - 13| > 13). ---- as if X - 13 > 14, then |X - 13| > 13

Now P(|X - 13| > 13) can be solved using chebychev inequality i.e.,

So P(|X - 13| > 13) = P(|X - 13| > = 14)

In our case k*sigma = 14, so k = 14/7

So P(|X - 13| > 13) <= 0.5^2 = 0.25

So it is atmost 0.25