Question

An electrical utility company offers reduced rates to homeowners who have installed “peak hours” meters. These meters effectively shut off high-consumption electrical appliances during the peak electrical usage hours. The utility company wants to inspect a sample of these meters to determine the proportion that is not working, either because they were bypassed or because of equipment failure. There are 45,300 meters installed and the utility does not want to inspect them all. An initial pilot sample of size 100 indicated that 14 meters were not working.

a.What additional sample is needed if the utility wants to estimate the proportion of meters that are not working with 95% confidence and maximum error of 3%.

b.A combined sample of size 800 indicated that 138 meters were not working properly. Find the 95% and 99% confidence intervals for the true proportion of the meters that are not working. Also interpret the 95% confidence interval.

Answer 1

(a)

Data given is as follows:

Significance level, a = 0.05

Margin of error, ME = 0.03

Sample size, n = 100

Sample proportion p = 14/100 = 0.14

For 95% CI, the margin of error is given by the formula:

ME = 1.96*SE

Here, SE = Standard error = (p*(1-p)/n)^0.5

Put the values:

0.03 = 1.96*(0.14*(1-0.14)/n)^0.5

Solving we get:

n = 513.9 = 514

So, additional sample needed = 514-100 = 414 more samples

(b)

In this case,

Sample size, n = 800

Sample proportion, p' = 138/800 = 0.173

The 95% CI is:

p' - 1.96*(p'*(1-p')/n)^0.5 < p < p' + 1.96*(p'*(1-p')/n)^0.5

0.173 - 1.96*(0.173*(1-0.173)/800)^0.5 < p < 0.173 + 1.96*(0.173*(1-0.173)/800)^0.5

0.147 < p < 0.199

The 99% CI is:

p' - 2.57*(p'*(1-p')/n)^0.5 < p < p' + 2.57*(p'*(1-p')/n)^0.5

0.173 - 2.57*(0.173*(1-0.173)/800)^0.5 < p < 0.173 + 2.57*(0.173*(1-0.173)/800)^0.5

0.139 < p < 0.207

Interpretation of 95% CI is:

We are 95% confident that the true population means lies within the interval of 0.147 to 0.199