Question

Engine parts A random sample of 16 of the more than $200$auto engine crankshafts produced in one day was selected. Here are measurements (in millimeters) of a critical component on these crankshafts:

a. Construct and interpret a $95\%$confidence interval for the mean length of this component on all the crankshafts produced on that day.

b. The mean length is supposed to be $\mu =224$mm but can drift away from this target during production. Does your interval from part (a) suggest that the mean has drifted from $224$mm? Explain your answer.

Short answer

Part a) The confidence interval is $(223.971,224.0.33)$

Part b) No, it has not drifted from$24mm$

Step-by-step solution

Part a) Step 1: Given information

The data set is

Part a) Step 2: The objective is to find out the 95%confidence interval for the mean length of the component.

We know,

The formula to compute the $t$- confidence interval for the population mean is:

$CI=\overline{x}\pm t_{\alpha /2,df}\times \frac{s}{\sqrt{n}}$

Because the population standard deviation is unknown, the $t$-confidence interval should be used.

The obtained excel output is:

Therefore, the required confidence interval is$(223.971,224.033)$

Part b) Step 1: The objective is to explain whether the mean has drifted from 224 mm 

The confidence interval is known from the preceding$(223.971,224.033)$ Since the upper limit is$224.033$, which is greater than$224$. As a result, the mean cannot be said to have deviated from $224mm$