# Hypothesis test question

A company considers buying a machine to manufacture a certain item. When tested, 28 out of 600 items produced by the machine were found defective. Does the data support the hypothesis that the defect rate of the machine is smaller than 3% at the 5% significance level.

$H_0: p \ge 0.03$

$H_a: p < 0.03$

$\alpha = 0.05$

Critical value $z = -1.645$

Reject $H_0$ if $z < -1.645$

Test statistic

$\Rightarrow Z_0 = \frac{\overline{X} - \mu_o}{\sigma/\sqrt{n}} = > \frac{0.0467-0.03}{0.03*0.97/\sqrt{600}} = 14.057$

Therefore, we fail to reject $H_0$. There isn't enough evidence to say that defect rate of machine is smaller than 3%.

Thanks!

• But you have not stated the value of $\sigma$, nor whether you know the population value, or if you obtained $\sigma$ from sample data, in which case your statistic would be a $t-$statistic.
– BFD
Oct 30 '13 at 6:53
• Sorry, I just mean that you did not mention $\sigma$ in the statement of the problem.
– BFD
Oct 30 '13 at 7:13
• BFD:Proportions hyposthesis testing, you don't need a sigma Oct 30 '13 at 9:34

H0: p=.03

H1: p<.03

p0=.03

p-hat = .0467 = 28/600

Test statistic => z = (0.0467 - 0.03)/sqrt(0.03*.97/600) = 2.398

z>-1.65 (One tailed test with alpha = .05)

Hence fail to reject H0 to conclude that there is not sufficient evidence that the defect

rate is smaller than 3%

Sorry, I changed the solution