Question

An initial time study resulted in an average observed time of 2.2 minutes per cycle, and a standard deviation of 0.3 minutes per cycle. The performance rating was 1.20. What sample size, including the 20 observations in the initial study, would be necessary to have a confidence of 95.44 percent that the observed time was within 4 percent of the true value?

Answer 1

Answer:

The sample size required is 225.

Step-by-step explanation:

The confidence interval for population mean (μ) is:

$\bar x\pm z_{\alpha /2}\times \frac{\sigma}{\sqrt{n}}$

The margin of error is:

$MOE= z_{\alpha /2}\times \frac{\sigma}{\sqrt{n}}$

Given:

$\bar x=2.2\\\sigma=0.3$

The margin of error is, MOE = 0.04

The confidence level is 95.44%.

The critical value of z for 95.44% confidence interval is:

$P(-2\leq Z\leq 2)=0.9544$

So, $z_{\alpha /2}=2$.

Determine the sample size as follows:

$MOE= z_{\alpha /2}\times \frac{\sigma}{\sqrt{n}}\\0.04=2\times\frac{0.30}{\sqrt{n}}\\ n=(15)^{2}\\=225$

Thus, the sample size required is 225.