Question

select the correct answer.
the transportation department of frederick inc. wanted to know the mode of transport used by their employees to commute to work. the survey was conducted by randomly asking 150 employees on a day when the factory had an attendance of 1,528 employees. the survey reported that 50% of those surveyed used public transportation to commute to work. assuming a 95% confidence level, which statement holds true?

a. as the sample size is too small, the margin of error is 0.079.
b. as the sample size is too small, the margin of error cannot be trusted.
c. as the sample size is appropriately large, the margin of error is 0.094.
d. as the sample size is appropriately large, the margin of error is 0.079.

Explanation:

Step1: Check sample - size criteria

A sample size of \(n = 150\) is considered reasonably large for estimating a proportion in many cases. The population size is \(N=1528\).

Step2: Use margin - of - error formula for proportion

The formula for the margin of error \(E\) for a proportion in a simple random sample (when the population is large or infinite - we can use the formula for infinite population here since \(n/N=150/1528\approx0.098\) and the finite - population correction factor is close to 1) at a 95% confidence level is \(E = z\sqrt{\frac{p(1 - p)}{n}}\), where for a 95% confidence level, \(z = 1.96\), and \(p = 0.56\) (since 56% of the surveyed used public transportation). First, calculate \((1 - p)=1 - 0.56 = 0.44\). Then \(\frac{p(1 - p)}{n}=\frac{0.56\times0.44}{150}=\frac{0.2464}{150}\approx0.001643\). And \(E=1.96\sqrt{0.001643}\approx1.96\times0.0405\approx0.079\).

Answer:

D. As the sample size is appropriately large, the margin of error is 0.079.