Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Solutions for Session 1, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6| B7 | B8| B9

Problem B1

Answers will vary. Here is one sample data set and the solutions to problems (a)-(e).

 Measurement Instrument Room Length (in Inches) Ruler 196 193 186 197 196 Yardstick 195 195 193 192 196
 a. The five measurements obtained using the ruler are not all exactly the same. The most likely cause of these differences is the method of measurement, such as the way the ruler was laid out. b. The five measurements obtained using the yardstick are not all exactly the same. Again, the most likely cause of these differences is the method of measurement. c. Yes, the answers are roughly similar. They should not be significantly different from one another, since the measurements were made in a similar way each time. However, the answers from the two measurement tools are not identical, since the methods of measurement were different. d. We would expect the yardstick to be more accurate than the ruler, since, with fewer measurements to be taken and added, there is less potential for error. e. We would expect a tape measure to be even more accurate than a yardstick, since there are still fewer measurements to take.

Problem B2

 a. You should find that the heights and arm spans are different for the six people, since people are inherently different and come in all shapes and sizes. b. No, they would probably not be identical. There are many potential reasons -- most likely, because of measurement errors, such as recording errors.

Problem B3

 a. The heights are not the same, nor are the arm spans. These measurements vary partly because of the differences between people. You may notice that we also have data on sex; these values vary depending on whether the person measured is male or female. The sex of an individual also has an effect on the variation in the list of heights and arm spans. Also, there is always a possibility of variation due to measurement errors. We cannot expect measurements of height or arm span with a meter stick to be exact every time. And still another type of measurement error may occur: A mistake might be made in recording the person's sex. b. Although the data suggest that men are typically taller than women, there is not enough data to prove this conclusively.

Problem B4

a.

Here are some possible sources of the variation in this data:

 • Measurement errors may have occurred. • Older pennies may be more worn than newer ones and therefore weigh less. • Different ingredients may have been used for making pennies in different years. • Pennies may have been made at different mints, using different equipment. • A penny may have something attached to it, such as a piece of dirt or gum.

b.

You should still expect to find some slight variation in the data as a result of measurement errors, but the values should be much closer than if you had measured 32 different pennies.

 Problem B5 We might expect the 33rd penny to weigh roughly three grams. There is no way to be absolutely sure of its weight beforehand, since there is variation in the data we were given for the first 32 pennies. Judging from the first 32 pennies, it is quite likely that the 33rd will be between 2.42 and 3.18 grams, and less likely that it will be between 3.00 and 3.10 grams.

 Problem B6 One possible source of the variation in this data is that some people are better at judging time than others; some may consistently overestimate or underestimate the minute. A second source, as always, is measurement error. Finally, people learn from experience -- their own or someone else's. After witnessing the measurement errors in the first estimate, they are likely to adjust their second estimates accordingly.

Problem B7

a.

Here are some possible sources of the variation in this data:

 • Measurement errors may have occurred. • Raisins come in many sizes and boxes are filled by weight. It takes fewer large raisins and more smaller raisins to fill a half-ounce box. • The machine that fills the boxes is not perfect; it may include too many or too few raisins. • Each box probably doesn't contain exactly one half ounce of raisins when you account for clumping or air in the box.

b.

Since all of the values were very close, and there are few enough possible values for the number of raisins, we should expect some of them to be exactly the same. This would also be true of a large class taking a test with 20 questions; some students in this class would get the same score, just by chance.

Problem B8

a.

Here are some possible sources of the variation in this data:

 • Differences of opinion • Measurement errors • Misread or misunderstood questions • Untruthful responses

b.

It may be that so many people would say "Yes" to this question that finding one of 25 people to say "No" is unlikely. Possibly a person may have misunderstood the question or may not have responded truthfully to what he or she perceived as a "loaded" question. Although the data suggest that another 25 people might respond in the same way, there is not enough data to prove this conclusively.

c.

The wording on each of these questions makes one think about the negative aspects of nuclear power. A respondant might be influenced to answer "Yes" to the final question after reading the first three. This questionnaire may be considered to be biased against nuclear power.

Problem B9

Here are the variables for each problem:

 • Problem B3: Sex, height, arm span • Problem B6: Time in seconds (i.e., people's estimates of 60 seconds) • Problem B7: Number of raisins in a box • Problem B8: Answers to Questions 1, 2, 3, and 4