Question

The U.S. Department of Agriculture claims that the mean consumption of bottled water by a person in the United States is 28.5 gallons per year. You believe that a person consume more than 28.5 gallons in bottled water per year. A random sample of 100 people in the United States has a mean bottled water consumption of 27.8 gallons per year with a standard deviation of 4.1 gallons. At α = 0.10 significance level can you reject the claim?

Answer 1

Answer: No

Step-by-step explanation:

Let $\mu$ be the population mean .

As per given , we have to test the hypothesis.

$H_0:\mu=28.5\\\\ H_a:\mu>28.5$

∵ Alternative hypothesis ($H_a$) is right-tailed , so our test is a right-tailed test.

Also, the population standard deviation is unknown to be 0.8 , so we use t-test.

Test statistic:$t=\dfrac{\overline{x}-\mu}{\dfrac{s}{\sqrt{n}}}$

, where $\overline{x}$ = Sample mean

$\mu$ = population mean

$s$ = sample standard deviation.

n= Sample size

Substitute $\overline{x}=27.8$

$s=4.1$

n=  100 , we get

$t=\dfrac{27.8-28.5}{\dfrac{4.1}{\sqrt{100}}}$

$t=\dfrac{-0.7}{\dfrac{4.1}{\sqrt{10}}}\approx-1.71$

By t-distribution, the critical t-value for degree of freedom 99 ( df =n-1) and significance level 0.10  :

$t_{\alpha,df}=1.29$

Decision : ∵ Calculated -value (-1.71) <  Critical value (1.29).

It means we do not reject the null hypothesis.

Conclusion : We do not have sufficient evidence at α = 0.10 significance level to reject the claim that the mean consumption of bottled water by a person in the United States is 28.5 gallons per year