Question

1. the caffeine content of large, brewed coffees sold at a local coffee shop follow a normal distribution, with a mean of 280 milligrams and a standard deviation of 22 milligrams. the caffeine content in phoebe’s coffee falls at the point in this distribution above which only about one - quarter of all large, brewed coffees fall. which one of the following statements is the most accurate description of the caffeine content of phoebe’s coffee? a. phoebe’s coffee contains approximately 295 milligrams of caffeine. b. phoebe’s coffee contains approximately 302 milligrams of caffeine. c. phoebe’s coffee contains approximately 210 milligrams of caffeine. d. phoebe’s coffee contains approximately 280 milligrams of caffeine. e. phoebe’s coffee contains approximately 258 milligrams of caffeine.

Explanation:

Step1: Identify the z - score

We know that the point above which only about one - quarter (0.25) of the data lies corresponds to a cumulative probability of \(1 - 0.25=0.75\) in the standard normal distribution. Looking up this value in the standard normal (z - table), the z - score \(z\) corresponding to a cumulative probability of 0.75 is approximately \(z = 0.67\).

Step2: Use the z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the value from the original normal distribution, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We know \(\mu = 280\), \(\sigma=22\), and \(z = 0.67\). Rearranging the formula for \(x\) gives \(x=\mu+z\sigma\).

Step3: Calculate the value of \(x\)

Substitute the values into the formula: \(x = 280+0.67\times22\). First, calculate \(0.67\times22 = 0.67\times22=14.74\). Then \(x=280 + 14.74=294.74\approx295\) milligrams.

Answer:

A. Phoebe's coffee contains approximately 295 milligrams of caffeine.