Question

American energy review reported that 27% of american households burn wood. if a random sample of 500 american households is selected find the mean, variance, and standard deviation of the number of households that burn wood.

Answer 1

For a random sample of 500 American households with 27% burning wood, the mean is 135 households, the variance is 98.55 households, and the standard deviation is approximately 9.9271 households.

Step-by-step explanation:

we use the binomial distribution because each household either does or does not burn wood (two outcomes). The probability of a household burning wood is given as 0.27 (or 27%).

Mean

The mean (expected value) of a binomial distribution is calculated using the formula:

Mean = n * p

Where n is the sample size, and p is the probability of success (in this case, the probability that a household burns wood).

Mean = 500 * 0.27 = 135 households

Variance

The variance of a binomial distribution is found using the formula:

Variance = n * p * (1 - p)

Variance = 500 * 0.27 * (1 - 0.27) = 98.55 households

Standard Deviation

The standard deviation is the square root of the variance:

Standard Deviation = sqrt(Variance) = sqrt(98.55) ≈ 9.9271 households

Answer 2

This is binomial distribution problem.
We are given that:

n = sample size = 500

p = proportion which burns wood = 0.27,

q = proportion which does not burn wood = 1-p = 0.73

A. Mean is calculated as:

Mean = n*p

Mean = 500 * 0.27

Mean = 135

B. Variance is calculated as:

Variance = n*p*q

Variance = 500*0.27*0.73

Variance = 98.55

C. Standard deviation is calculated as:

Standard deviation = sqrt(variance)

Standard deviation = sqrt(98.55)

Standard deviation = 9.93