Question
Asked Oct 31, 2019

From a sample with n=32, the mean number of televisions per household is 4 with a standard deviation of 1 television. Using Chebychev's theorem, determine at least how many of the households have between 2 and 6 televisions.

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Expert Answer

Step 1

Chebyshev’s rule can be used for the large set data. It says that at least 1-1/k2 percentage of the data falls within the k standard deviation of the mean, where k is a positive number greater than 1. It can be represented as:

1
P(-ka<x<ko) >1-
k2
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1 P(-ka<x<ko) >1- k2

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Step 2

According to the provided data, for a sample size, n is 32, the mean, µ is 4 with a standard deviation, σ of 1.

The number of standard deviations under which the values 2 and 6 are falling can be obtained by calculating the z-score.

2-4
1
=-2
6-4
=2
Therefore, it can be said that the values between 2 and 6 are all within 2
standard deviations of the mean. Hence, the value of k is 2
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2-4 1 =-2 6-4 =2 Therefore, it can be said that the values between 2 and 6 are all within 2 standard deviations of the mean. Hence, the value of k is 2

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Step 3

Using Chebyshev’s Inequality, the percenta...

Ρ(μ- ko<x<μ+ kσ) >1
k2
P(2<X<6) >1-
22
P(2<X <6)>1-
4
P(2<X< 6)>0.75
P(2<X<6)> 75%
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Ρ(μ- ko<x<μ+ kσ) >1 k2 P(2<X<6) >1- 22 P(2<X <6)>1- 4 P(2<X< 6)>0.75 P(2<X<6)> 75%

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