The mathematical goal of this activity is for students to use the area under a normal curve to find the proportion of values in certain intervals. While the values in this activity are chosen to highlight the usual areas within 1, 2, and 3 standard deviations of the mean, it is not essential for students in this course to memorize those values.

In the digital version of the activity, students use technology to find the area under a normal curve within an interval. The technology allows students to find more precise approximations of the area. Use the digital version if students would benefit from focusing on the meaning of the area rather than on calculations.

The goal of this discussion is for students to understand that a normal curve can be used to estimate the proportion of data that lies within a certain interval when the data is approximately normal. Ask:

• “Why is it important that we are told that the lifespans are approximately normally distributed?” (If the data had a different type of distribution, modeling with a normal distribution would not accurately describe the situation. Compare this to modeling quadratic data with a line.)
• “Why are the answers for the first 4 questions the same?” (Since the normal curve is determined by the standard deviation and mean, the area to the left of the position 1 standard deviation below the mean is the same, regardless of the values of the mean and standard deviation. Since the normal curve is symmetric, the area less than 1 standard deviation below the mean is equal to the area greater than 1 standard deviation above the mean.)
• “Since all normal curves have the same areas for the regions relative to the mean and number of standard deviations away, approximately what percentage of values are within 1 standard deviation of the mean?” (about 68%)
• “Approximately what percentage of values are within 2 standard deviations of the mean?” (about 95%)
• “Approximately what percentage of values are within 3 standard deviations of the mean?” (about 99.7%)

The mathematical purpose of this activity is to use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate the interval given various proportions. Estimating an interval given various proportions is not required by the standards, and students will not be assessed on questions like the ones in this activity, but these types of questions are included here so students have the opportunity to understand more thoroughly the concept of using the area under a normal curve to find the proportion of values in certain intervals.

In the digital version of the activity, students use technology to find areas under a normal curve. The technology allows students to find the area quickly and accurately. Use the digital version if students would benefit from focusing on the meaning of the area rather than on calculations.

The purpose of this discussion is for students to make connections between the area under the normal curve and the wait times at a restaurant. Here are some questions for discussion.

• “What percentage of customers wait within 1 standard deviation of the mean (from 21.1 to 27.5 minutes)?” (approximately 68%)
• “Approximately how long do the middle 95% of the customers wait?” (They wait within 2 standard deviations of the mean, from 17.9 to 30.7 minutes.)
• “A second restaurant has a longer mean wait time of 36.5 minutes and a standard deviation of 10.2 minutes. About how long do you think the middle 50% of customers wait at the second restaurant? How could you check your results? Explain your reasoning.” (I think they wait between a little more than 26.3 minutes and a little less than 46.7 minutes. I know that about 68% of the customers wait between 26.3 and 46.7 minutes because that is within 1 standard deviation. I could get the actual times by modeling this with technology.)
• “Why is a normal curve a useful way to model wait times?” (The model is useful because the same model could be used to measure wait times at a variety of restaurants. Changing the mean wait time or the variability, as measured by the standard deviation, does not impact how the model is used.)

If students struggle to find the correct estimates, consider asking:  

• “Is this a question about LED or incandescent bulbs? Clearly mark each question to show if it is related to LED or incandescent bulbs.”   

• “Which values for the mean and standard deviation should you use for each situation? Where would you enter them into your technology?”