Answer:
Since the pvalue of 0.0985 > 0.05, we cannot infer that the satisfaction rate is less than the claim with a level of significance of 5%
Step-by-step explanation:
The null hypothesis is:
![H_{0} = 0.9](https://tex.z-dn.net/?f=H_%7B0%7D%20%3D%200.9)
Because tate Farm claims that policyholders have a customer satisfaction rate of greater than 90%.
The alternate hypotesis is:
![H_{1} < 0.9](https://tex.z-dn.net/?f=H_%7B1%7D%20%3C%200.9)
Because of the question: Can we infer that the satisfaction rate is less than the claim with a level of significance of 5%.
The test statistic is:
![z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
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.
State Farm claims that policyholders have a customer satisfaction rate of greater than 90%.
This means that ![\mu = 0.9, \sigma = \sqrt{0.9*0.1}](https://tex.z-dn.net/?f=%5Cmu%20%3D%200.9%2C%20%5Csigma%20%3D%20%5Csqrt%7B0.9%2A0.1%7D)
To check the accuracy of this claim, a random sample of 60 State Farm policyholders was asked to rate whether they were satisfied with the quality of customer service.
This means that ![n = 60](https://tex.z-dn.net/?f=n%20%3D%2060)
Fifteen percent of these policyholders said they were not satisfied with the quality of service.
So 100 - 15 = 85% were satisfied, which means that ![p = 0.85](https://tex.z-dn.net/?f=p%20%3D%200.85)
Test statistic:
![z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![z = \frac{0.85 - 0.9}{\frac{\sqrt{0.9*0.1}}{\sqrt{60}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B0.85%20-%200.9%7D%7B%5Cfrac%7B%5Csqrt%7B0.9%2A0.1%7D%7D%7B%5Csqrt%7B60%7D%7D%7D)
![z = -1.29](https://tex.z-dn.net/?f=z%20%3D%20-1.29)
has a pvalue of 0.0985.
Since the pvalue of 0.0985 > 0.05, we cannot infer that the satisfaction rate is less than the claim with a level of significance of 5%