Answer:
a) <u>Alternative hypothesis:</u> the use of the coupons is isgnificantly higher than 10%.
<u>Null hypothesis:</u> the use of the coupons is not significantly higher than 10%.
The null and alternative hypothesis can be written as:
![H_0: \pi=0.1\\\\H_a:\pi>0.1](https://tex.z-dn.net/?f=H_0%3A%20%5Cpi%3D0.1%5C%5C%5C%5CH_a%3A%5Cpi%3E0.1)
b) Point estimate p=0.13
c) At a significance level of 0.05, there is not enough evidence to support the claim that the proportion of coupons use is significantly higher than 10%.
Eagle should not go national with the promotion as there is no evidence it has been succesful.
Step-by-step explanation:
<em>The question is incomplete.</em>
<em>The sample data shows that x=13 out of n=100 use the coupons.</em>
This is a hypothesis test for a proportion.
The claim is that the proportion of coupons use is significantly higher than 10%.
Then, the null and alternative hypothesis are:
![H_0: \pi=0.1\\\\H_a:\pi>0.1](https://tex.z-dn.net/?f=H_0%3A%20%5Cpi%3D0.1%5C%5C%5C%5CH_a%3A%5Cpi%3E0.1)
The significance level is 0.05.
The sample has a size n=100.
The point estimate for the population proportion is the sample proportion and has a value of p=0.13.
![p=X/n=13/100=0.13](https://tex.z-dn.net/?f=p%3DX%2Fn%3D13%2F100%3D0.13)
The standard error of the proportion is:
![\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.1*0.9}{100}}\\\\\\ \sigma_p=\sqrt{0.0009}=0.03](https://tex.z-dn.net/?f=%5Csigma_p%3D%5Csqrt%7B%5Cdfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B0.1%2A0.9%7D%7B100%7D%7D%5C%5C%5C%5C%5C%5C%20%5Csigma_p%3D%5Csqrt%7B0.0009%7D%3D0.03)
Then, we can calculate the z-statistic as:
![z=\dfrac{p-\pi-0.5/n}{\sigma_p}=\dfrac{0.13-0.1-0.5/100}{0.03}=\dfrac{0.025}{0.03}=0.833](https://tex.z-dn.net/?f=z%3D%5Cdfrac%7Bp-%5Cpi-0.5%2Fn%7D%7B%5Csigma_p%7D%3D%5Cdfrac%7B0.13-0.1-0.5%2F100%7D%7B0.03%7D%3D%5Cdfrac%7B0.025%7D%7B0.03%7D%3D0.833)
This test is a right-tailed test, so the P-value for this test is calculated as:
As the P-value (0.202) is greater than the significance level (0.05), the effect is not significant.
The null hypothesis failed to be rejected.
At a significance level of 0.05, there is not enough evidence to support the claim that the proportion of coupons use is significantly higher than 10%.