Question

1. the empirical rule (or the 68 - 95 - 99.7 rule) tells us that approximately 68% of the pumpkin spice lattes in this distribution have serving temperatures between 152 degrees fahrenheit and 164 degrees fahrenheit.
2. the empirical rule tells us that approximately 95% of the pumpkin spice lattes in this distribution have serving temperatures between 146 degrees fahrenheit and 170 degrees fahrenheit.
3. the empirical rule tells us that approximately 99.7% of the pumpkin spice lattes in this distribution have serving temperatures between 140 degrees fahrenheit and 176 degrees fahrenheit.
4. true or false? according to the empirical rule, approximately 16% of the pumpkin spice lattes in this distribution have serving temperatures that are hotter than 164 degrees fahrenheit.

Explanation:

Step1: Recall the Empirical Rule

The Empirical Rule for a normal - distribution states that about 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$), about 95% lies within 2 standard deviations ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard deviations ($\mu\pm3\sigma$). The normal distribution is symmetric about the mean. So, if 68% of the data is in the interval $(\mu - \sigma,\mu+\sigma)$, then the proportion of data outside this interval is $1 - 0.68=0.32$, and the proportion of data in each tail is $\frac{0.32}{2}=0.16$.

Step2: Analyze the 68 - 95 - 99.7 Rule for the given problem

For the 68% interval, we have temperatures from 152 to 164. For the 95% interval, from 146 to 170. For the 99.7% interval, from 140 to 176. Since the distribution is symmetric about the mean, and 68% of the data is in the interval (152, 164), the proportion of data above 164 (the upper - tail of the 68% interval) is $\frac{1 - 0.68}{2}=0.16$ or 16%.

Answer:

True