Question

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. colby’s 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?
colby consumes about 2.3 times as much caffeine as the average college student
colby consumes about 275 milligrams more caffeine than the average college student
colby’s caffeine consumption is at the 23rd percentile.
colby consumes about 353 milligrams less caffeine than the average college student.
colby consumes about 138 milligrams more caffeine than the average college student.
question 8
1 pts
at one major hospital in a large city, administrators keep track of the amount of time, in minutes, that patients wait in the emergency room before being seen by a doctor. it turns out the distribution of wait times at this hospital follows a symmetric bell - shaped pattern, with a mean of 42 minutes and a standard deviation of 9 minutes. isaac’s wait time is 30 minutes. which one of the following statements about isaac’s wait time is most accurate?
isaac’s wait time is unusual compared to other patients’ wait times.
isaac’s wait time is at about the 90th percentile, and this means that only about 10% of patients waited longer than isaac.
isaac’s wait time is at about the 10th percentile, and this means that only about 10% of patients waited for less time than isaac.
isaac’s wait time is at about the 10th percentile, and this means that only about 10% of patients waited longer than isaac.
isaac’s wait time is at about the 90th percentile, and this means that only about 10% of patients waited for less time than isaac.

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.

Step2: Solve for Colby's caffeine consumption

We know that $z = 2.3$, $\mu=215$ and $\sigma = 60$. Rearranging the z - score formula to solve for $x$ gives $x=\mu+z\sigma$. Substituting the values, we get $x=215 + 2.3\times60=215+138 = 353$. The difference between Colby's consumption and the mean is $x-\mu=353 - 215=138$. So Colby consumes about 138 milligrams more caffeine than the average college student.

Step3: Calculate Isaac's z - score

For Isaac, $\mu = 42$, $\sigma=9$ and $x = 30$. Using the z - score formula $z=\frac{x-\mu}{\sigma}=\frac{30 - 42}{9}=\frac{- 12}{9}\approx - 1.33$. Looking up the z - score in the standard normal distribution table, a z - score of approximately - 1.33 corresponds to the 9th - 10th percentile. This means that about 10% of patients waited for less time than Isaac.

Answer:

Colby consumes about 138 milligrams more caffeine than the average college student
Isaac's wait time is at about the 10th percentile, and this means that only about 10% of patients waited for less time than Isaac