Question

Lasers can provide highly accurate measurements of small movements. To determine the accuracy of such a laser, it was used to take 82 measurements of a known quantity. The sample mean error was 20 μm with a standard deviation of 60 μm. The laser is properly calibrated if the mean error is μ = 0. A test is made of H0 : μ = 0 versus H1 : μ ≠ 0. Find the P value?

Answer 1

Answer: $p-value \approx 0.01$

Step-by-step explanation:

Hypothesis testing

$\\\left\{\begin{matrix}H_0 : \mu_0 =0\\ H_1: \mu_0 \neq 0\end{matrix}\right.$

For this problem, we need to use the t-student distribution to  make inference about the data. We calculate the t-statistics as below:

$\bar{X} = 20\:\mu m\\S.E. = 60\:\mu m\\n=82\\t_{stat} = \frac{\bar{X}-\mu_0}{S.E/\sqrt{n}}=\frac{20-0}{60/\sqrt{82}}=3.0185$

Using a t-statistics table, or using the function TDIST in Microsoft Excel with $t_{stat} = 3.0185,\: df (degree\: of\: freedom) = 14 - 1 = 13$ with two-tailed distribution, we obtain $p-value \approx 0.01$.