Question

D STARBKS 14.8 Caffeine in Starbucks coffee. Scientists at the University of Florida College of Medicine investigated the level of caffeine in 16-ounce cups of Starbucks coffee (Journal of Analytical Toxicology, Oct. 2003). In one phase of the experiment, cups of Starbucks Breakfast Blend (a mix of Latin American coffees) were purchased on six consecutive days from a single specialty coffee shop. The amount of caffeine in each of the six cups ( measured in milligrams) is provided in the following table. 564 498 259 303 300 307

1. Suppose the scientists are interested in determining whether the median amount of caffeine in Breakfast Blend coffee exceeds 300 milligrams. Set up the null and alternative hypotheses of interest.

2. How many of the cups in the sample have a caffeine content that exceeds 300 milligrams?

3. Suppose your answer in (b) is x. Find the p-value using pbinom in R. Note: the p-value here is P (X ≥ x), where X ∼ Bin(6, 0.5). What is your conclusion?

Answer 1

Hi,

Here we have to determine whether the median amount of caffeine in breakfast blend coffee exceeds 300 milligrams.

Setting up the null and alternative hypothesis,

Null hypothesis: The median amount of caffeine in breakfast blend coffee is equal to 300 milligrams.

Alternative hypothesis: The median amount of caffeine in breakfast blend coffee is greater than 300 milligrams.

Arrange all the observations in increasing order

259 300 303 307 498 564

Median = (3rd observation + 4th observation)/2

Median of sample = (303 + 307)/2

Median = 305

4 cups in the sample have a caffeine content that exceeds 300milligrams.

Use lower.tail = FALSE if we want the probability of values x or larger. P[X >= x]

> pbinom(4, size = 6, prob = 0.5, lower.tail = FALSE)
[1] 0.109375

P-value = 0.109375

Here, p-value is greater than level of significance 0.05 hence we accept nul hypothesis.

Hence we conclude that the median amount of caffeine in breakfast blend coffee is equal to 300 milligrams.