Question

A researcher wishes to estimate the proportion of X-ray machines that malfunction. A random 275 sample of machines is taken, and 228 of the machines in the sample malfunction.
Based upon this, compute a 95% confidence interval for the proportion of all X-ray machines that malfunction. Then complete the table below.
Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places.
What is the lower limit of the 95% confidence interval?
What is the upper limit of the 95% confidence interval?

Answer 1

Answer:

Following are the solution to the given question:

Step-by-step explanation:

$95\%$ Confidence Interval for both the percentage of all x-ray machines

p = the machinery's share is not working:

$= \frac{228}{275}\\\\ = 0.829$

$\text{Margin of Error} = Z_{(\frac{\alpha}{2})} \times \sqrt{( p \times (1-p)}{n})$

                         $= 1.96 \times \sqrt{(0.829 \times \frac{0.171}{275})} \\\\= 1.96 \times 0.023 \\\\= 0.045$

Lower $95\%$ Confidence interval = p - error margin $= 0.829 - 0.045 = 0.784$

Upper $95\%$ Confidence Interval = p + error margin$= 0.829 + 0.045 = 0.874$

So, $95\%$ Confidence Interval $= ( 0.78 , 0.87 )$