Respuesta :
Answer:
We can't conclude that the return rate is less than 20%.
Step-by-step explanation:
We have a sample of size n=6985 and a success rate of p=1322/6985=0.1892.
We have to test the claim that the return rate is less than p=0.2, so the null and alternative hypothesis are:
[tex]H_0: \pi\geq0.2\\\\H_1: \pi<0.2[/tex]
The significance level is 0.01.
The standard deviation for this sample is
[tex]\sigma=\sqrt{\frac{\pi(1-\pi)}{N} }= \sqrt{\frac{0.2(1-0.2)}{6985}}=0.0048[/tex]
Then, the test statistic is
[tex]z=\frac{p-\pi}{\sigma} =\frac{0.1892-0.2}{0.0048}= -2.237[/tex]
We look up the P-value in a z-table for z=-2.237.
The P-value is 0.01264.
This P-value is greater than the significance level, so the effect is not significant and the null hypothesis can't be rejected.
We can't conclude that the return rate is less than 20%.
Testing the hypothesis for the proportion, it can be concluded that since the p-value of the test is 0.0126 > 0.01, which means that we cannot conclude that the return rate is less than 20%.
At the null hypothesis, we test if the return rate is of at least 20%, that is:
[tex]H_0: p \geq 0.2[/tex]
At the alternative hypothesis, we test if the return rate is of less than 20%, that is:
[tex]H_1: p < 0.2[/tex]
Using the normal approximation to the binomial, the test statistic is given by:
[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
In which:
- [tex]\overline{p}[/tex] is the sample proportion.
- p is the proportion tested at the null hypothesis.
- n is the sample size.
For this problem, the parameters are:
[tex]p = 0.2, n = 6985, \overline{p} = \frac{1322}{6985} = 0.1893[/tex]
The value of the test statistic is:
[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
[tex]z = \frac{0.1893 - 0.2}{\sqrt{\frac{0.2(0.8)}{6985}}}[/tex]
[tex]z = -2.24[/tex]
The p-value of the test is the probability of finding a sample proportion of 0.1893 or below, which is the p-value of z = -2.24.
Looking at the z-table, z = -2.24 has a p-value of 0.0126.
The p-value of the test is 0.0126 > 0.01, which means that we cannot conclude that the return rate is less than 20%.
A similar problem is given at https://brainly.com/question/24166849
