Question

A machinist is making engine parts with axle diameter of 0.7 cm. A random sample of 10 parts shows mean diameter 0.742 cm with a standard deviation of 0.04 cm. On the basis of this sample, find if you would say that the work is inferior.
(Given $t_9(0.05) = 2.262$)

Hint:

In hypothesis testing, reject $H_0$ if the computed test statistic exceeds the critical value from the $t$-distribution table.

Answer 1

We perform a one-sample $t$-test to compare sample mean with population mean.
Let the null hypothesis $H_0$: the work is not inferior, i.e., $\mu = 0.7$
Sample mean $\bar{x} = 0.742$
Population mean $\mu = 0.7$
Sample size $n = 10$
Standard deviation $s = 0.04$
Test statistic is calculated as:
\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{0.742 - 0.7}{0.04 / \sqrt{10}} = \frac{0.042}{0.01265} \approx 3.32 \]
Now compare this with the critical value $t_9(0.05) = 2.262$
Since calculated $t = 3.32>2.262$, we reject $H_0$.
This means the sample mean is significantly greater than the target diameter.
Hence, we conclude that the work is inferior.