Intent Students apply their understanding of the normal distribution and standard deviation to situations involving variability.

Mathematics This activity asks students to reason from the properties of a normal distribution. Given the mean and standard deviation of several sets of data, they will be asked to reason about the amount of data that reside in various regions under the normal curve.

Progression Students will work on this activity individually and then share their results in a class discussion.

Approximate Time 5 minutes for introduction 25 minutes for activity (at home or in class) 20 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Materials Transparencies of the graphs of the data (optional) [link to BLM pdf of Can Your Calculator Pass…, p. 8]

Doing the Activity Suggest that students draw normal distributions, and label the mean and two standard deviations on either side of the mean, for each situation. Then they can justify their thinking using their curves and the expected percentages. If students need a reminder about the percentages related to a normal distribution, refer them back to the Standard Deviation Basics reference pages.

Discussing and Debriefing the Activity For Question 1, students should use the fact that in a normal distribution, approximately 68% of all results occur within one standard deviation of the mean. Therefore, about 68% of the bottles should pass inspection, while 32% will need to be removed for correction. You might paraphrase Question 1 as follows: Suppose a bottle is chosen at random before the quality-control inspection. What is the probability that it ought to be removed? Students should recognize that this is essentially the same question they just answered and that the probability is .32. To lead into Question 2, ask, How is Question 2 different from Question 1? Question 2 is set up in an opposite manner to Question 1. That is, Question 1 provides boundary or cutoff values and asks for a percentage, whereas Question 2 gives the percentage and asks for a cutoff value. Specifically, the manufacturer wants to set a cutoff time so that only 2.5% of calculators will need repair before that time. Students will probably use a diagram of the appropriate normal curve to illustrate what this cutoff time should be. Question 2 is complicated by the fact that it involves only one “tail” of the normal distribution. The horizontal axis of the diagram below shows the mean as well as the values one and two standard deviations above and below the mean. The shaded portions represent results that are at least two standard deviations from the mean.

If students understand the percentages for standard deviation, they will see that the two shaded areas together total 5% of all calculators. Thus, the calculators that break down before 819 days constitute about 2.5% of the total. (Approximately 95% will break down somewhere between the day 819 and day 1151; another 2.5% will last more than 1151 days.) In other words, if the manufacturer offers to replace calculators that break in less than 819 days, about 2.5% will need replacement. Question 3 combines features of the first two questions. Like Question 1, it provides the cutoff value and asks for a percentage. Like Question 2, it is a “one-tail” problem, concerned only with the percentage of students whose scores are above a certain value. It is complicated by the fact that the difference between the cutoff and the mean is not a whole-number multiple of the standard deviation. Again, a diagram is probably useful. In the diagram below, the shaded region is the area between the mean and one standard deviation above the mean. Students should be able to see that this area represents about 34% of all results. Thus, about 16% of students get scores above 610, and the percentage getting scores above 600 is slightly higher.

Don’t get bogged down by the fine points of estimating the percentage of scores above 600. Although this is an interesting question, it is not the intended focus. You can mention that there are tables that provide details on areas like these so that people don’t have to make visual estimates every time they come across a problem of this type.

Key Question How is Question 2 different from Question 1?

Supplemental Activity More About Soft Drinks, Calculators, and Tests (reinforcement or extension) involves an asymmetric “tail” problem (beyond two standard deviations at one end, beyond one standard deviation at the other end). Questions 2 and 3 require students to estimate areas under the normal curve whose boundaries are not whole multiples of the standard deviation from the mean. [Link to calculator instructions for statistics on the TI-84]