Answer:
The mean of C is 170 households
The standard deviation of C, is approximately 5 households
Step-by-step explanation:
The given parameters are;
The percentage of households in the United States that had a computer in 2014 = 85%
The size of the randomly selected sample in 2014, n = 200
The random variable representing the number of households that had a computer = C
Therefore, we have;
The probability of a household having a computer P = 85/100 = 0.85
Let
Therefore;
The mean (expected) number in the sample, μₓ, = E(x) = n × P is given as follows;
μₓ = 200 × 0.85 = 170
The mean of C = μₓ = 170
The variance, σ² = n × P × (1 - P) = 200 × 0.85 × (1 - 0.85) = 25.5
Therefore;
The standard deviation, σ = √(σ²) = √(25.5) ≈ 5.05
The standard deviation of C, σ ≈ 5 households (we round (down) to the nearest whole number)
The mean and the standard deviation of C are 170 and 5.05 respectively
The given parameters are:
[tex]\mathbf{n = 200}[/tex] -- the sample size
[tex]\mathbf{p = 85\%}[/tex] -- the proportion of household that had a computer
(a) The mean
This is calculated as:
[tex]\mathbf{\bar x = np}[/tex]
So, we have:
[tex]\mathbf{\bar x = 200 \times 85\%}[/tex]
[tex]\mathbf{\bar x = 170}[/tex]
(b) The standard deviation
This is calculated as:
[tex]\mathbf{\sigma = \sqrt{np(1 - p)}}[/tex]
So, we have:
[tex]\mathbf{\sigma = \sqrt{170 \times (1 - 85\%)}}[/tex]
[tex]\mathbf{\sigma = \sqrt{170 \times 15\%}}[/tex]
[tex]\mathbf{\sigma = \sqrt{25.5}}[/tex]
Take square roots
[tex]\mathbf{\sigma = 5.05}[/tex]
Hence, the mean and the standard deviation of C are 170 and 5.05 respectively
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