Question

a rainstorm in portland, oregon, wiped out the electricity in 10% of the households in the city. suppose that a random sample of 80 portland households is taken after the rainstorm.

answer the following.

(if necessary, consult a list of formulas.)

(a) estimate the number of households in the sample that lost electricity by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). do not round your response.

(b) quantify the uncertainty of your estimate by giving the standard deviation of the distribution. round your response to at least three decimal places.

Explanation:

Identify the distribution parameters

We model the number of households that lost electricity, \(X\), using a Binomial Distribution.

• The sample size is \(n = 80\).
• The probability of a household losing electricity is \(p = 10\% = 0.1\).

Calculate the expected value

Using the Binomial Mean concept, the expectation of a binomial random variable is:

$$ \mu = n \cdot p $$

Substituting the values:

$$ \mu = 80 \cdot 0.1 = 8 $$

Calculate the standard deviation

Using the Binomial Standard Deviation concept, the standard deviation of a binomial random variable is:

$$ \sigma = \sqrt{n \cdot p \cdot (1 - p)} $$

Substituting the values:

$$ \sigma = \sqrt{80 \cdot 0.1 \cdot 0.9} = \sqrt{7.2} $$

Compute the final numerical value

We evaluate the square root:

$$ \sigma \approx 2.68328 $$

Rounding to at least three decimal places gives \(2.683\).

Answer:

(a) Estimate the number of households in the sample that lost electricity by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response.
<blank>8</blank>

(b) Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.
<blank>2.683</blank>