You obtain a sample from a normal distribution with variance 2 = 2.0 and µ = 1.0

Question

You obtain a sample from a normal distribution with variance 2 = 2.0 and µ = 1.0.

(a) (5 points) Use Chebychev’s inequality to provide a bound for the probability that an observation from this distribution is larger than 7 (take advantage of the fact that the normal distribution is symmetric around the mean).

(b) (5 points) Use the normal distribution to calculate this probability exactly with two decimals precision.

(c) (5 points) Use Chebychev’s inequality to determine the sample size, n, needed to ensure that the probability that the sample mean, xn, is between 0 and 2, is at least 0.9.

Explanation / Answer

a)

here k score =(X-mean)/std deviation

for x=7 ; k =(7-1)/(2)1/2 =6/(2)1/2

from Chebychev’s inequality  probability that an observation from this distribution is larger than 7 =1/(2k2)

=1/(2*(6/(2)1/2)2) =1/36 =0.02778

b)

c)

let sample size =n

for std error of mean =std deviaiton/(n)1/2

therefore from  Chebychev’s inequality  probability that the sample mean, xbar, is between 0 and 2

=P(0<X<2) =P(-1<|X-1|<1) =1-1/k2 >0.9

k=(X-1)/std error =(2-1)/(2/n)1/2 =(n/2)1/2

1-1/(n/2) >0.9

1-2/n >0.9

2/n =<1-0.9

2/n =<0.1

n>=2/0.1

n>=20

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