Question

A coffee shop claims that its fresh-brewed drinks have a mean caffeine content of 80 milligrams per 5 ounces. A city health agency believes that the coffee shop’s fresh- brewed drinks have higher caffeine content. To test this claim the health agency takes a random sample of 100 five-ounce servings and found the average mean caffeine content of the sample was 87 milligrams with standard deviation of 25 milligrams. Does this provide enough evidence at the 1% significance level to claim that the coffee shop’s fresh- brewed drinks have higher caffeine content?

For the test of significance questions, clearly indicate each of the formal steps in the test of significance.

Step 1: State the null and alternative hypothesis.

Step 2: Calculate the test statistic.

Step 3: Find the p-value.

Step 4: State your conclusion. (Do not just say “Reject H0” or “Do not reject H0”, state the conclusion in the context of the problem.)

Answer 1

1) H₀: = 80

   H₁: > 80

2) The test statistic t = ()/(s/)

                             = (87 - 80)/(25/)

                             = 2.8

3) P-value = P(T > 2.8)

                  = 1 - P(T < 2.8)

                  = 1 - 0.9969

                  = 0.0031

Since the P-value is less thn the significance level (0.0031 < 0.01), so we should reject H₀.

So at 1% significance level there is sufficient evidence to support the claim that the coffee shop's fresh-brewed drinks have higher caffeine content.