Answer:
The pvalue of the test is 0.0007 < 0.01, which means that we reject the null hypothesis and accept the alternate hypothesis that the percentage of defective items produced by this machine is greater than 5%.
Step-by-step explanation:
A machine, when working properly, produces 5% or less defective items.
This means that the null hypothesis is:
Test if the percentage of defective items produced by this machine is greater than 5%.
This means that the alternate hypothesis is:
The test statistic is:
In which X is the sample mean, 
is the value tested at the null hypothesis, 
is the standard deviation and n is the size of the sample.
0.05 is tested at the null hypothesis:
This means that 
A random sample of 300 items taken from the production line contained 27 defective items.
This means that 
Value of the test statistic:
Pvalue of the test:
Testing if the mean is greater than a value, which means that the pvalue of the test is 1 subtracted by the pvalue of Z = 3.18, which is the probability of a finding a sample proportion of 0.09 or higher.
Looking at the Z-table, Z = 3.18 has a pvalue of 0.9993
1 - 0.9993 = 0.0007
The pvalue of the test is 0.0007 < 0.01, which means that we reject the null hypothesis and accept the alternate hypothesis that the percentage of defective items produced by this machine is greater than 5%.
![F(\infty)= \int\limits^{\infty}_{-\infty} {f(y)} \, dy=1 \\ \\ \Rightarrow \int\limits^1_0 {cy} \, dy +\int\limits^2_1 {(2-y)} \, dy=1 \\ \\ \Rightarrow \left. \frac{cy^2}{2} \right]^1_0+\left[2y- \frac{y^2}{2} \right]^2_1=1 \\ \\ \Rightarrow \frac{c}{2} +4-2-2+ \frac{1}{2} =1 \\ \\ \Rightarrow \frac{c}{2} = \frac{1}{2} \\ \\ \Rightarrow \bold{c=1}](https://tex.z-dn.net/?f=F%28%5Cinfty%29%3D%20%5Cint%5Climits%5E%7B%5Cinfty%7D_%7B-%5Cinfty%7D%20%7Bf%28y%29%7D%20%5C%2C%20dy%3D1%20%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20%5Cint%5Climits%5E1_0%20%7Bcy%7D%20%5C%2C%20dy%20%2B%5Cint%5Climits%5E2_1%20%7B%282-y%29%7D%20%5C%2C%20dy%3D1%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20%5Cleft.%20%5Cfrac%7Bcy%5E2%7D%7B2%7D%20%5Cright%5D%5E1_0%2B%5Cleft%5B2y-%20%5Cfrac%7By%5E2%7D%7B2%7D%20%5Cright%5D%5E2_1%3D1%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20%20%5Cfrac%7Bc%7D%7B2%7D%20%2B4-2-2%2B%20%5Cfrac%7B1%7D%7B2%7D%20%3D1%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20%5Cfrac%7Bc%7D%7B2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20%5Cbold%7Bc%3D1%7D%20)
![E(y)=\int\limits^{\infty}_{-\infty} {yf(y)} \, dy \\ \\ =\int\limits^1_0 {y^2} \, dy +\int\limits^2_1 {(2y-y^2)} \, dy \\ \\ =\left. \frac{y^3}{3} \right]^1_0+\left[y^2- \frac{y^3}{3} \right]^2_1 \\ \\ = \frac{1}{3} +4- \frac{8}{3} -1+ \frac{1}{3} \\ \\ =1](https://tex.z-dn.net/?f=E%28y%29%3D%5Cint%5Climits%5E%7B%5Cinfty%7D_%7B-%5Cinfty%7D%20%7Byf%28y%29%7D%20%5C%2C%20dy%20%5C%5C%20%20%5C%5C%20%3D%5Cint%5Climits%5E1_0%20%7By%5E2%7D%20%5C%2C%20dy%20%2B%5Cint%5Climits%5E2_1%20%7B%282y-y%5E2%29%7D%20%5C%2C%20dy%20%5C%5C%20%20%5C%5C%20%3D%5Cleft.%20%5Cfrac%7By%5E3%7D%7B3%7D%20%5Cright%5D%5E1_0%2B%5Cleft%5By%5E2-%20%5Cfrac%7By%5E3%7D%7B3%7D%20%5Cright%5D%5E2_1%20%5C%5C%20%20%5C%5C%20%3D%20%5Cfrac%7B1%7D%7B3%7D%20%2B4-%20%5Cfrac%7B8%7D%7B3%7D%20-1%2B%20%5Cfrac%7B1%7D%7B3%7D%20%20%5C%5C%20%20%5C%5C%20%3D1)