Question

the battery life of a new smartphone model is normally distributed, with a mean of 15.2 hours and a standard deviation of 1.3 hours. according to the empirical rule, approximately 95% of these smartphones have battery lives that are between hours and hours. please do not round your answers when you enter them into the above textboxes! question 3 1 pts 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 phoebes 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 phoebes coffee? phoebes coffee contains approximately 295 milligrams of caffeine. phoebes coffee contains approximately 302 milligrams of caffeine. phoebes coffee contains approximately 210 milligrams of caffeine. phoebes coffee contains approximately 280 milligrams of caffeine. phoebes coffee contains approximately 258 milligrams of caffeine.

Explanation:

Step1: Recall the Empirical Rule for normal distribution

The Empirical Rule states that for a normal - distributed data, approximately 95% of the data lies within 2 standard deviations of the mean.

Step2: Calculate the lower bound

The mean $\mu = 15.2$ hours and the standard deviation $\sigma=1.3$ hours. The lower bound is $\mu - 2\sigma=15.2-2\times1.3=15.2 - 2.6 = 12.6$ hours.

Step3: Calculate the upper bound

The upper bound is $\mu + 2\sigma=15.2 + 2\times1.3=15.2+2.6 = 17.8$ hours.

Step4: For the caffeine - content problem

If Phoebe's coffee is at the point above which only about one - quarter of all large, brewed coffees fall, we are looking for the 75th percentile of the normal distribution. The z - score corresponding to the 75th percentile is approximately $z = 0.674$. Using the formula $x=\mu+z\sigma$, where $\mu = 280$ milligrams and $\sigma = 22$ milligrams. Then $x=280+0.674\times22=280 + 14.828\approx295$ milligrams.

Answer:

For the first part: 12.6, 17.8
For the second part: Phoebe's coffee contains approximately 295 milligrams of caffeine.