A machine produces washers of thickness 0.50 mm. To determine whether the machine is in proper working order, a sample of 10 washers is chosen for which the mean thickness is 0.53 mm and the standard deviation is 0.03 mm. Test the hypothesis at 5% level of significance that the machine is working in proper order. (Given $t_{9 (0.05)} = 2.262$)

No, the machine is not working properly; the deviation is statistically significant.

The correct answer is Option (2) → No, the machine is not working properly; the deviation is statistically significant.

Here, $μ_0 = 0.50$

Given $μ_0 = 0.50 mm, \bar x = 0.53 mm, s = 0.03 mm$ and $n = 10d.f. = 10-1=9$

$t =\frac{\bar x - μ_0}{\frac{S}{\sqrt{n}}}=\frac{0.53-0.50}{0.03}×\sqrt{10}=3.16$

Let the hypothesis be given as

Null hypothesis $H_0$: There is no significant difference between $\bar x$ and $μ_0$.

Alternative hypothesis $H_a$: There is significant difference between $\bar x$ and $μ_0$.

Given that level of significance $= 5\% ⇒α = 0.05$

Also, $t_{d.f (α)} = t_{9 (0.05)} = 2.262$

We have calculated $t = 3.16$

$∵ t>t_α$, so, null hypothesis is rejected.

Hence, the machine is not working in proper order.