Answer:
Explained below.
Step-by-step explanation:
(a)
In this case we need to determine if the proportion of customer complaints has decreased by more than 10% under the new packaging design.
The hypothesis can be defined as follows:
<em>H</em>₀: The proportion of customer complaints has not decreased by more than 10% under the new packaging design, i.e. <em>p</em>₁ - <em>p</em>₂ ≤ 0.10.
<em>Hₐ</em>: The proportion of customer complaints has decreased by more than 10% under the new packaging design, i.e. <em>p</em>₁ - <em>p</em>₂ > 0.10.
(b)
The information provided is:
n₁ = 220
n₂ = 220
X₁ = 86
X₂ = 41
Compute the sample proportions and total proportions as follows:
![\hat p_{1}=\frac{X_{1}}{n_{1}}=\frac{86}{220}=0.3909\\\\\hat p_{2}=\frac{X_{2}}{n_{2}}=\frac{41}{220}=0.1864\\\\\hat P=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{86+41}{220+220}=0.2886](https://tex.z-dn.net/?f=%5Chat%20p_%7B1%7D%3D%5Cfrac%7BX_%7B1%7D%7D%7Bn_%7B1%7D%7D%3D%5Cfrac%7B86%7D%7B220%7D%3D0.3909%5C%5C%5C%5C%5Chat%20p_%7B2%7D%3D%5Cfrac%7BX_%7B2%7D%7D%7Bn_%7B2%7D%7D%3D%5Cfrac%7B41%7D%7B220%7D%3D0.1864%5C%5C%5C%5C%5Chat%20P%3D%5Cfrac%7BX_%7B1%7D%2BX_%7B2%7D%7D%7Bn_%7B1%7D%2Bn_%7B2%7D%7D%3D%5Cfrac%7B86%2B41%7D%7B220%2B220%7D%3D0.2886)
Compute the test statistic value as follows:
![z=\frac{\hat p_{1}-\hat p_{2}}{\sqrt{\hat P(1-\hat P)[\frac{1}{n_{1}}+\frac{1}{n_{2}}]}}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B%5Chat%20p_%7B1%7D-%5Chat%20p_%7B2%7D%7D%7B%5Csqrt%7B%5Chat%20P%281-%5Chat%20P%29%5B%5Cfrac%7B1%7D%7Bn_%7B1%7D%7D%2B%5Cfrac%7B1%7D%7Bn_%7B2%7D%7D%5D%7D%7D)
![=\frac{0.3909-0.1864}{\sqrt{0.2886(1-0.2886)\times[\frac{1}{220}+\frac{1}{220}]}}\\\\=\frac{0.2045}{\sqrt{0.0018665}}\\\\=4.733524\\\\\approx 4.73](https://tex.z-dn.net/?f=%3D%5Cfrac%7B0.3909-0.1864%7D%7B%5Csqrt%7B0.2886%281-0.2886%29%5Ctimes%5B%5Cfrac%7B1%7D%7B220%7D%2B%5Cfrac%7B1%7D%7B220%7D%5D%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B0.2045%7D%7B%5Csqrt%7B0.0018665%7D%7D%5C%5C%5C%5C%3D4.733524%5C%5C%5C%5C%5Capprox%204.73)
The test statistic value is 4.73.
Compute the <em>p</em>-value as follows:
![p-value=P(Z>4.73)](https://tex.z-dn.net/?f=p-value%3DP%28Z%3E4.73%29)
![=1-P(Z](https://tex.z-dn.net/?f=%3D1-P%28Z%3C4.73%29%5C%5C%3D1-%28%5Capprox%201%29%5C%5C%3D0)
The <em>p</em>-value of the test is 0.
(c)
The decision rule is:
The null hypothesis will be rejected if the p-value of the test is less than the significance level.
<em>p</em>-value = 0 < <em>α</em> = 0.01.
The null hypothesis will be rejected at 1%significance level.
Thus, there is enough evidence at 1% significance level to support the engineers' claim.
(d)
The decision rule is:
The null hypothesis will be rejected if the p-value of the test is less than the significance level.
<em>p</em>-value = 0 < <em>α</em> = 0.10.
The null hypothesis will be rejected at 10%significance level.
Thus, there is enough evidence at 10% significance level to support the engineers' claim.