Answer:
The proportion of defective chips produced by the machine is more than 4% so the machine needs an adjustment.
Step-by-step explanation:
In this case we need to test whether the proportion of defective chips produced by the machine is more than 4%.
The hypothesis can be defined as follows:
<em>H</em>₀: The proportion of defective chips produced by the machine is not more than 4%, i.e. <em>p</em> ≤ 0.04.
<em>H</em>ₐ: The proportion of defective chips produced by the machine is more than 4%, i.e. <em>p</em> > 0.04.
The information provided is:
<em>X</em> = 12
<em>n</em> = 200
<em>α</em> = 0.025
The sample proportion of defective chips is:
![\hat p=\frac{X}{n}\\\\=\frac{12}{200}\\\\=0.06](https://tex.z-dn.net/?f=%5Chat%20p%3D%5Cfrac%7BX%7D%7Bn%7D%5C%5C%5C%5C%3D%5Cfrac%7B12%7D%7B200%7D%5C%5C%5C%5C%3D0.06)
Compute the test statistic as follows:
![z=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}\\\\=\frac{0.06-0.04}{\sqrt{\frac{0.04(1-0.04)}{200}}}\\\\=1.44](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B%5Chat%20p-p%7D%7B%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B0.06-0.04%7D%7B%5Csqrt%7B%5Cfrac%7B0.04%281-0.04%29%7D%7B200%7D%7D%7D%5C%5C%5C%5C%3D1.44)
The test statistic value is 1.44.
Decision rule:
We reject a hypothesis if the p-value of a statistic is lower than the level of significance α.
Compute the <em>p</em>-value of the test:
![p-value=P(Z>1.44)\\=1-P(Z](https://tex.z-dn.net/?f=p-value%3DP%28Z%3E1.44%29%5C%5C%3D1-P%28Z%3C1.44%29%5C%5C%3D1-0.92507%5C%5C%3D0.07493%5C%5C%5Capprox%200.075)
The <em>p</em>-value of the test is 0.075.
<em>p</em>-value = 0.075 > <em>α</em> = 0.025
The null hypothesis was failed to be rejected at 2.5% level of significance.
Thus, it can be concluded that the proportion of defective chips produced by the machine is more than 4% so the machine needs an adjustment.