Ten cartons are taken at random from an automatic packing machine. The mean net weight of the ten cartons is 11.8 kg and standard deviation is 0.15 kg. Does the sample mean differ significantly from the intended mean of 12 kg?
[Given that for $d.f. = 9, t_{0.05} = 2.26$]
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Yes, the sample mean differs significantly from the intended mean.
Given $μ_0 = 12$ kg, $\bar x = 11.8$ kg, $s = 0.15$ and $n = 10⇒ d.f. = n − 1 = 9$
$t =\frac{\bar x - μ_0}{\frac{s}{\sqrt{n}}}=\frac{11.8-12}{\frac{0.15}{\sqrt{10}}}=-4.216$
Let the hypothesis be given as
Null hypothesis $H_0$ = There is no significant difference between $\bar x$ and $μ$.
Alternate hypothesis $H_a$ = There is significant difference between $\bar x$ and $μ$.
Given that $t_α = t_{0.05} = 2.26$ at $d.f. = 9$
We have calculated $t = -4.216$
$∵t≤-t_α$. So, null hypothesis is reject.
Hence, sample mean differ significantly from the intended mean of 12 kg.