Answer to Question #148374 in Statistics and Probability for Haris Ahad

Question

From past experience, a professor knows that the test score of a student taking her finalexam is a continuous random variable with meanμ= 75 and varianceσ2= 25.„(a) Use Markov’s inequality to give an upper bound for the probability that a student’s testscore will exceed 85.(b) Use Chebyshev’s inequality to determine what can be said about the probability that astudent will score between 65 and 85.(c) How many students would have to take the exam to ensure, with probability at least0.95, that the classaveragewould be within 2 marks of 75? Assume that you can applythe Central Limit Theorem.(d) How accurate do you think Markov’s inequality and Chebyshev’s inequality are? Do youthink it’s worth using them in practice? Why or why not?

(d) Suppose you are handed a data sample from some statistical survey. How would you determine whether the sample comes from a distribution with this density?

Answer 1

 1)Use Markov’s inequality. P(X > 85) ≤ E[X]/85 = 75/85 = 15/17

2) Here we use Chebyshev. P(65 ≤ X ≤ 85) = P(|X − 75| ≤ 10) = 1 − P(|X − 75| > 10) ≥ 1 − var(X)/10² = 1 − 25/100 = 3/4

3) P(|X −75| ≤ 5) = 1−P(|X −75| > 5) ≥ 1−var(X)/25 = 0.9. So 1/n = 0.1 and so n = 10

4)  P(|X − 75| ≤ 5) = P(|Z| ≤ $\sqrt n$ ) = 0.9 and so Φ($\sqrt n$ ) = 0.95. A normal table lookup gives $\sqrt n$ = 1.65 and so n ≈ 3. This is a situation where you should not trust the answer, since for n = 3 a CLT approximation is not valid anyway. However the Chebyshev is valid and gives a correct but possibly not tight answer

d)we assume that the data collected by the polling company is pretty representative of the population at large