Question

1. the amount of caffeine consumed daily by college students follows a normal distribution, with a mean of 215 milligrams and a standard deviation of 60 milligrams. colbys caffeine consumption, when converted to a z - score, is equal to 2.3. based on this information, which one of the following statements is correct? a. colby consumes about 2.3 times as much caffeine as the average college student. b. colby consumes about 275 milligrams more caffeine than the average college student. c. colbys caffeine consumption is at the 23rd percentile. d. colby consumes about 353 milligrams less caffeine than the average college student. e. colby consumes about 138 milligrams more caffeine than the average college student.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know that $z = 2.3$, $\mu=215$ and $\sigma = 60$, and we want to find $x$.

Step2: Rearrange the formula to solve for $x$

From $z=\frac{x - \mu}{\sigma}$, we can get $x=\mu+z\sigma$.

Step3: Substitute the given values

Substitute $\mu = 215$, $z = 2.3$ and $\sigma=60$ into the formula: $x=215+2.3\times60$.
$x=215 + 138=353$. The difference between Colby's consumption ($x$) and the mean ($\mu$) is $x-\mu=353 - 215=138$.

Answer:

E. Colby consumes about 138 milligrams more caffeine than the average college student.