Question

a company that makes cola drinks states that the mean caffeine content per 12 - ounce bottle of cola is 65 milligrams. you want to test this claim. during your test, you find that a random sample of thirty 12 - ounce bottles of cola has a mean caffeine content of 64.5 milligrams. assume the population is normally distributed and the population standard deviation is 1.1 milligrams. at α = 0.03, can you reject the companys claim? complete parts (a) through (e).
(c) find the standardized test statistic.
z = - 0.36 (round to two decimal places as needed.)
(d) decide whether to reject or fail to reject the null hypothesis.
a. since z is in the rejection region, fail to reject the null hypothesis.
b. since z is in the rejection region, 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 3% significance level, there enough evidence to the companys claim that the mean caffeine content per 12 - ounce bottle of cola milligrams.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0:\mu = 65\) (the company's claim) and the alternative hypothesis \(H_1:\mu
eq65\) (two - tailed test).

Step2: Calculate the z - score

The formula for the z - score is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). Given \(\bar{x} = 44.5\), \(\mu=65\), \(\sigma = 7.7\), \(n = 30\).

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For a two - tailed test with \(\alpha=0.03\), the critical values are \(z_{\alpha/2}=\pm z_{0.015}\). From the standard normal table, \(z_{0.015}\approx\pm 2.17\)

Step3: Make a decision

Since \(z=-14.58\) and the critical values are \(z=\pm2.17\), and \(-14.58<-2.17\) (i.e., \(z\) is in the rejection region)

Answer:

(b) B. Since \(z\) is in the rejection region, reject the null hypothesis.
(a) At the \(3\%\) significance level, there is enough evidence to reject the company's claim that the mean caffeine content per \(12 -\)ounce bottle of cola is \(65\) milligrams.