Question

a company that makes cola drinks states that the mean caffeine content per 12 - ounce bottle of cola is 35 milligrams. you want to test this claim. during your tests, you find that a random sample of thirty 12 - ounce bottles of cola has a mean caffeine content of 33.9 milligrams. assume the population is normally distributed and the population standard deviation is 6.4 milligrams. at \\( \alpha=0.02 \\), can you reject the companys claim? complete parts (a) through (e).
a. since \\( z \\) is in the rejection region, reject the null hypothesis.
b. since \\( z \\) is in the rejection region, fail to reject the null hypothesis.
c. since \\( z \\) is not in the rejection region, fail to reject the null hypothesis.
d. since \\( z \\) is not in the rejection region, reject the null hypothesis.
(e) interpret the decision in the context of the original claim.
at the \\( 2 \\% \\) significance level, there is not enough evidence to support the companys claim that the mean caffeine content per 12 - ounce bottle of cola is equal to 35 milligrams.

Explanation:

Step1: Calculate the test statistic \( z \)

The formula for the \( z \)-test statistic is \( z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}} \), where \( \bar{x} = 33.9 \), \( \mu = 35 \), \( \sigma = 6.4 \), and \( n = 30 \).

$$ z=\frac{33.9 - 35}{\frac{6.4}{\sqrt{30}}}=\frac{- 1.1}{\frac{6.4}{5.477}}\approx\frac{-1.1}{1.168}\approx - 0.94 $$

Step2: Determine the critical values

For a two - tailed test with \( \alpha=0.02 \), the critical values are \( z_{\alpha/2}=\pm z_{0.01} \). From the standard normal table, \( z_{0.01} = 2.33 \), so the critical values are \( z=-2.33 \) and \( z = 2.33 \).

Step3: Check the position of the test statistic

Since \( - 2.33<-0.94<2.33 \), the \( z \)-value is not in the rejection region.

Answer:

C. Since \( z \) is not in the rejection region, fail to reject the null hypothesis.