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

A one-sample $t$-test was conducted to assess if the sample mean differs from the population mean. The null hypothesis, $H_0$, posited that the work is not inferior, meaning $\mu = 0.7$. The sample mean ($\bar{x}$) was 0.742, with a population mean ($\mu$) of 0.7 and a sample size ($n$) of 10. The sample standard deviation ($s$) was 0.04. The test statistic was computed as follows:
\[t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{0.742 - 0.7}{0.04 / \sqrt{10}} = \frac{0.042}{0.01265} \approx 3.32\]
This calculated $t$-statistic of 3.32 was compared to the critical value $t_9(0.05) = 2.262$. Since $3.32>2.262$, the null hypothesis $H_0$ was rejected. This indicates that the sample mean is significantly greater than the target diameter. Consequently, the work is concluded to be inferior.