Question

part 1: practice with the normal distribution
the serving temperatures of pumpkin spice lattes at a local coffee shop are normally distributed, with a mean temperature of 158 degrees fahrenheit and a standard deviation of 6 degrees fahrenheit. please use this information to answer questions 1 through 15.
hint: remember that a very useful first step in these types of problems involves drawing the distribution and marking it out three standard deviations on either side of the mean. to help you get started, weve shared a normal curve with you below. please mark it accordingly, if you wish, as you work through the first part of this assignment.

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. true

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 of the mean ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard deviations of the mean ($\mu\pm3\sigma$). Given $\mu = 158$ and $\sigma=6$.

Step2: Calculate for 68% interval

For the 68% interval ($\mu\pm\sigma$), we have $158 - 6=152$ and $158 + 6 = 164$.

Step3: Calculate for 95% interval

For the 95% interval ($\mu\pm2\sigma$), we calculate $158-2\times6=158 - 12 = 146$ and $158+2\times6=158 + 12 = 170$.

Step4: Calculate for 99.7% interval

For the 99.7% interval ($\mu\pm3\sigma$), we calculate $158-3\times6=158 - 18 = 140$ and $158+3\times6=158 + 18 = 176$.

Step5: Analyze the 16% statement

Since 68% of the data is within $\mu\pm\sigma$ (i.e., between 152 and 164), the remaining 32% is outside this interval. Half of this 32% (i.e., 16%) is above 164.

Answer:

1. 152, 164
2. 146, 170
3. 140, 176
4. True