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 ____ degrees fahrenheit and ____ degrees fahrenheit.
2. the empirical rule tells us that approximately 95% of the pumpkin spice lattes in this distribution have serving temperatures between ____ degrees fahrenheit and ____ degrees fahrenheit.
3. the empirical rule tells us that approximately 99.7% of the pumpkin spice lattes in this distribution have serving temperatures between ____ degrees fahrenheit and ____ 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, 95% within 2 standard - deviations, and 99.7% within 3 standard - deviations. Given mean $\mu = 158$ and standard deviation $\sigma=6$.

Step2: Calculate bounds for 68%

For 68% of the data, the lower bound is $\mu-\sigma=158 - 6=152$ and the upper bound is $\mu+\sigma=158 + 6=164$.

Step3: Calculate bounds for 95%

For 95% of the data, the lower bound is $\mu - 2\sigma=158-2\times6=158 - 12 = 146$ and the upper bound is $\mu+2\sigma=158 + 2\times6=158+12 = 170$.

Step4: Calculate bounds for 99.7%

For 99.7% of the data, the lower bound is $\mu-3\sigma=158-3\times6=158 - 18 = 140$ and the upper bound is $\mu+3\sigma=158+3\times6=158 + 18 = 176$.

Step5: Analyze the statement in question 4

The value 164 is $\mu+\sigma$. Since 68% of the data is between $\mu-\sigma$ and $\mu+\sigma$, the percentage of data above $\mu+\sigma$ is $\frac{100 - 68}{2}=16\%$.

Answer:

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