Show your work. No work-No credit. 6. (16 points) A group of university faculty

Question

Show your work. No work-No credit. 6. (16 points) A group of university faculty at a large state university rated the job basket ball coach on a scale of o to 10 with higher numbers indicating greater satisfaction. (Faculty were allowed to give fractional ratings, so you may consider the rating to be a continuous The average rating was 5 and the standard deviation was 1. Assume that the distribution of responses (a) (4 points) According to the empirical rule, approximatelyperceat of the responses must have been less than 4 (b) (4 points) According to the empirical rule, approximately 95 percent of the responses must have been between and __ -, (e) (4 points) According to Chebyshev's rule, at lease_percent of the responses must have been between 3 and 7. d) (4 points) According to Chebyshev's rule, at least 8/9 proportion of the responses must have been between and .

Explanation / Answer

a)
P(X < 4)
= P(X < 5 -1)
= P(X < mean - 1* sd)
= 0.16

b)
95 % lies between two sd within mean
= (5 - 2* 1 , 5 + 2*1 )
= (3,7)

c)
by chebyshev's inequality
P(|X - 5| > 2 * 1) <= 1/ 2^2
= P(|X-5| < 2) >= 1 - 1/4
P(3 < X< 7) >= 0.75

d)

P(|X-5| < 1*k) >= 1 - 1/k^2
k = 3
hence
P(|X-5| < 3) >= 1 - 1/3^2
P(2 < X< 8) >= 8/9 or 89%
hence
between 2 and 8

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