Answer to Question #222353 in Statistics and Probability for janz

Question

1.    A manufacturer packs sugar into plastic bags. Each bag is to hold 5 kilos of sugar. When the production process is under control, each bag contains an average of 5 kilos. At one period, a sample of 17 bags was taken to check the process and was found to weigh 5.6 kilos with standard deviation of 0.75 kilos. Is the manufacturing process under control? (N.B If the sample mean deviates significantly from 5 kilos, the process is no under control)

Answer 1

$\bar{x}=5.6 \\ s=0.75 \\ n=17 \\ H_0: \mu=5 \\ H_1: \mu ≠5$

n<30 and $\sigma^2$ is unknown

Test-statistic:

$t= \frac{\bar{x}-\mu}{s/ \sqrt{n}} \\ t= \frac{5.6-5}{0.75 / \sqrt{17}} \\ = \frac{0.6}{0.182} \\ = 3.296$

Let’s use 5 % significance level.

Two-tailed test.

Reject $H_0 \;if |t|\; > t_{α/2, n-1}$

α=0.05

d.f. = n-1=17-1=16

$t_{α/2, n-1} = t_{0.025, 16}=2.12$

$|t|>t_{tab}$

Reject $H_0$ .

Therefore, we conclude that there is sufficient evidence that the production process is NO under control at 5 % significance level.