Which Toothpaste is Most Popular? 2

TEACHER EDITION

List of Activities for this Unit:

ACTIVITY / STRAND / DESCRIPTION
1 – Which Toothpaste is Most Popular? / PS / Collecting Data: Limiting Bias
2 – What is Average? / PS / Measures of Central Tendency
3 – Is the Sample Biased / PS / Under-coverage Bias and Voluntary Response Bias
4 –Another Fish Story / PS / Estimating Total by Sampling – Simulation using M&Ms
5- Practice Problem (only one)
COE Connections / Another Deer Story
Homecoming Queen
MATERIALS / 1 large container
500 or more M&Ms or beads in 5 different colors
Additional M&Ms or beads in a 6th, different color
Small measuring tool that can hold 50 M&Ms or beads
Calculator
Graph Paper
Warm-Ups
(in Segmented Extras Folder)

Vocabulary: Mathematics and ELL

average / outliers
bias / quantitative
categorical / response bias
characterize / roster
comparisons / sample sizes
data / simulating
difficulties / survey
elapses / tag
elicit / trials
encounter / under-coverage Bias
frequency / visualize
generalize / voluntary response bias
mean

Essential Questions:

  • How is data used to make and support claims?
  • What is the difference between categorical data and quantitative data?
  • What is an average?
  • Why are averages used?
  • When is using an average appropriate or not appropriate?
  • What claims can be made about sets of data using measures of central tendency?
  • How can data from samples be used to determine information about the entire population?
  • When is using sampling data appropriate or not appropriate for determining data about an entire population?
  • What is meant by “bias”?
  • What are different kinds of bias?
  • What are different ways a question or survey could be biased?
  • How can bias be avoided in survey questions, collection methods, etc.?
  • How can you control for different kinds of bias?
  • What information does a set of data tell us?
  • What do measures of central tendency tell us about data sets?

Lesson Overview:

  • Before allowing the students the opportunity to start the activity: access their prior knowledge with regards to how data is collected, what data they have collected, what claims they have heard or seen on TV, radio, or the internet, what claims they have made based on data, what data they have used or seen used to support claims, etc.
  • Discuss different kinds of data that can be gathered using a survey. What data is categorical (descriptive with no natural order)and what data is quantitative (numerical)? Are there other kinds of data?
  • Discuss what can be done with sample data after it is gathered. What claims can be made using sample data or measures of central tendency?
  • When evaluating a claim, what data would you use to support or contradict the claim?
  • When describing a set of data, why might certain measures of central tendency be appropriate or not appropriate?
  • Given sample data, what conclusions can you make about the entire population? What does it mean if two sample data sets are not exactly the same? Is this a problem?
  • Use resources from your building.

Performance Expectations:

4.4.EDetermine the median, mode, and range of a set of data and describe what each measure indicates about the data.

5.5.BDetermine and interpret the mean of a small data set of whole numbers.

5.6.JMake and test conjectures based on data (or information) collected from explorations and experiments.

6.3.FDetermine the experimental probability of a simple event using data collected in an experiment.

6.6.HMake and test conjectures based on data (or information) collected from explorations and experiments.

7.4.ARepresent the sample space of probability experiments in multiple ways, including tree diagrams and organized lists.

7.4.CDescribe a data set using measures of center (median, mean, and mode) and variability (maximum, minimum, and range) and evaluate the suitability and limitations of using each measure for different situations.

7.6.HMake and test conjectures based on data (or information) collected from explorations and experiments.

8.3.ASummarize and compare data sets in terms of variability and measures of center.

8.3.DDescribe different methods of selecting statistical samples and analyze the strengths and weaknesses of each method.

8.5.HMake and test conjectures based on data (or information) collected from explorations and experiments.

A1.6.AUse and evaluate the accuracy of summary statistics to describe and compare data sets.

Performance Expectations and Aligned Problems

Chapter 24 “Which Toothpaste is Most Popular” Subsections: / 1-
Which Toothpaste is Most Popular? / 2-
What is Average? / 3-
Is the Sample Biased? / 4-
Another Fish Story. / 5-
Practice Problem
Problems Supporting:
PE 4.4.E ≈ 5.5.B ≈ 7.4.C ≈ 8.3.A ≈ A1.6.A / 2
(weak) / 8 – 10
(weak) / 19 – 21, 23
(weak) / 27, 29, 31
(weak)
Problems Supporting:
PE 5.6.J ≈ 6.3.F ≈ 6.6.H ≈ 7.6.H ≈ 8.5.H / 24 - 34
Problems Supporting:
PE 7.4.A / 27, 29
(weak)
Problems Supporting:
PE 8.3.D / 1 – 7
(background) / 15 - 23

Assessment: Use the multiple choice and short answer items from Probability and Statistics that are included in the CD. They can be used as formative and/or summative assessments attached to this lesson or later when the students are being given an overall summative assessment.

M&M/Bead Sampling Activity (page 17 teacher edition and page 12 student edition):

Teacher Directions to #4 in “Another Fish Story”

To do this activity, you will need at least one large container and a large number of M&Ms, beads, beans, or similar small objects. There should be about 500 of these objects, enough where counting all the objects to determine the number of individual objects in a subgroup of the entire population would not be feasible and sampling would be a more reasonable approach.

It is not necessary that you know the number of objects in each subgroup within the entire population. Through group sampling of the entire population, students can determine a reasonable number for the objects in each subset and have discussions around whether or not they believe the numbers that were determined are reasonable.

For the purposes of explaining the activity, we reference M&Ms, though any similar objects can be used. With approximately 4 pounds of M&Ms, sort out one color of M&M and set those aside. Place the rest of the M&Ms into a large container. Then, count out a set number of the color you sorted out from the rest. The number you choose will vary with the total number of M&Ms, but it could be about half of the number of that color that were in the entire population to start with. Keep track of this number because the students will need this number to determine estimates for the other colors of M&Ms. Place these M&Ms into the container and mix thoroughly.

Divide students into groups and give each group approximately 50 M&Ms from the thoroughly mixed container of all M&Ms. Have them create or fill in a chart to document the number of each color of M&M in their sample. Repeat this sampling process two more times. Each sample size should be approximately the same size.

Tell students the number of the special color that you sorted out from the rest and have students use this information to make estimates about the number of each other color in the entire population. Students can do this by using percents, proportional reasoning, or other appropriate methods that do not include counting the entire population.

Have students discuss their impression of the reasonableness of their estimates and how the sample size impacted their estimates.

WARNING EATING THE DATA PRIOR TO COMPLETING THIS ACTIVITY ALTERS THE OUTCOME.

1 - Which Toothpaste is Most Popular?

(Collecting Data: Limiting Bias)

What is the definition of survey? (To ask someone for information in order to collect data for the study of some aspect of a group or area is a survey.)

1. While watching TV the other night I listened to a commercial that claimed “four out of five dentists surveyed handed out ‘Smile Bright’ after dental visits”. Claims similar to this one are heard and/or read daily. Some data is used to provide information and other data is used to try to persuade.

a. List four types of data (information) an average (common)person might encounter (come across)on any given day. Discuss the purposes for which the data might be intended. (Think about different data you might see as a consumer, a parent, a teacher, a public official, a medical patient, a student, etc.) For example: A Trident gum commercial states that “four out of five dentists surveyed recommend sugarless gum for their patients who chew gum.” Television commercials attempt to persuade us to buy a certain product.

  1. 5 out of 4 people have problems with fractions______

______

______

  1. daily high temperatures in a newspaper______

______

______

  1. advertisements for sales of goods______

______

______

  1. sporting statistics ______

______

______

2. A frequencylist showingthe number of purchases of different flavors of ice cream provides information called categorical data. The categories are the flavors that were bought. ACategorical datalist is not the same as a list of the percent of nutritional characteristics of a flavor of ice cream; this nutritional data is called quantitativedata.

Describe the difference between categorical and quantitative data. ______

To be complete and meaningful, quantitative information consists of both quantitative data - the numbers - and categorical data - the labels that tell us what the numbers measure.______

The frequency is the number of times an item appears in a set of data.

Categorical data is data in categories like: food, clothes, and hobbies.

Quantitativedata is data that can be counted or measured.

3. Data was collected on the ages of male and female vocalists in the Grammy Awards. Comparisons(associations)were made to see if the ages for the genders that won Grammy’s were about the same age.

Is the data Categorical or Quantitative? Quantitative______

Explain the reasoning behind your decision. ______

We are comparing theages so we are using quantitative data.______

4. What type of graph(s) would you use to best display this type of data? Explain your choice(s).

If you want to emphasize individual items or to support discrete comparisons of multiple values at the same location along the interval scale, bar graphs work best.

Line graphs are used to emphasize the overall shape of the data or changes from one item to the next.

Because we are trying to make comparisons, the best type of graph to use in this instance is a bar graph.

5. A local do-it-yourself supply store asks for your zip code. The information providesthe store with information about the where their customers come from to buy do-it-yourself supplies.

Is the data Categorical or Quantitative? Categorical______

Explain the reasoning behind your decision. _(Answers will vary.)______

This data is categorical because it is identifying customers based on the region in which they are located. This can be slippery for kids and adults to “get” because zip codes are numbers and it is natural to think the data must be quantitative, but the numbers are merely a way to locate the place an area of land. Look at zip codes asa list or as a map. Zip codes identify cities or regions of large cities. The same is true for area codes for phone numbers and street addresses; they are categorical data.

6. What type of graph(s) would you use to best display this type of data? Explain your choice(s).

If you want to emphasize individual items or to support discrete comparisons of multiple values at the same location along the interval scale, bar graphs work best.

Line graphs are used to emphasize the overall shape of the data or changes from one item to the next.

Because we are trying to make comparisons of customer locations, the best type of graph to use in this instance is a bar graph.

7. A survey of models of students’ cars in the school parking lot was made at each school in the district. Explain how the use of the data collected would be classified as Categorical and how the use of the data would be classified as Quantitative.

Categorical_because of the classification/type/make of cars

Quantitative because of the number that represents each category

a. What type of graph(s) would you use to best display this type of the Categorical data? Explain your choice(s).

Because we are trying to make comparisons between the data, the best type of graph to use in this instance is a bar graph.

b. What type of graph(s) would you use to best display this type of the Quantitative data? Explain your choice(s).

Because we are trying to make comparisons between the data, the best type of graph to use in this instance is a bar graph; however, a line graph could also be appropriate because we can use this data to grasp an overall view of the information.

2 - What is Average?

  1. Why do you think we averagegrades?

An average is a quantitative measurement of the cumulative achievement of a student’s performance; it “irons out” the highs and the lows to show the students overall score.

An average is a measure of central tendency known as the mean. The average is the sum of the data values divided by the number of data values.

  1. Suppose a student has an average of 82% in one of his or her classes. What does that mean?

This means that they have earned 82 % of the total points possible in the class.

The data is a list of the heights (in inches) of the starting lineup of the Whitehouse High School boys’ basketball team.

74, 74, 74, 74, 74

a. Describe this team for someone who has not seen the data. Make three different statements about the basketball team that characterize (describe) the data. Include the mean in one of your statements and the median or mode in one of your statements.

  1. All of the players are 74 inches tall.
  1. The mean of the players’ heights is 74 inches.
  1. The mode (median) of the players’ heights is 74 inches.

b. What is it about this data set that simplifies this descriptive process? ______

All of the data points have the same value.

c. The following sets of data represent the heights, in inches, of the starting lineups of the Bullard and TroupHigh School boys’ basketball teams.

Bullard: 74, 73, 75, 75, 73

Troup: 68, 73, 76, 77, 76

Describe this team for someone who has not seen the data. Make three different statements about the basketball team that characterize (describe) the data. Use the mean, median and mode when making your statements about the two teams.

  1. The mean of the Bullard High School boys’ basketball team is 74 inches tall.

The mean of the Troup High School boys’ basketball team is 74 inches tall.

  1. The median of the Bullard High School boys’ basketball team is 74 inches tall.

The median of the Troup High School boys’ basketball team is 76 inches tall.

  1. The Bullard High School boys’ basketball team is bimodal: 73 and 75 inches tall.

The mode of the Troup High School boys’ basketball team is 76 inches tall.

d. Compare the statements you made about the different teams.

  1. One similarity:Their means are the same.______

______

  1. One difference:Their modes and medians are different.______

______

  1. During this task we have seen that the meanfor the Whitehouse, Bullard, and Troup data is equal.

Would you say this common value for the mean is equally representative of the three different

sets of data? _____Yes______Explain your reasoning. The average of the players’ heights from each of the schools’ starting lineups is representative of each team.

  1. Using your reasoning from abovegeneralize (simplify) as to where the idea of “averaging” is not appropriate?

If you are trying to find the height of the tallest member of each team, the idea of “averaging” (finding the mean) is not appropriate.

12. Suppose you are asked to summarize or characterize the set of data that represents the heights of the starting lineups of all high school basketball teams in the state of Washington, both public and private. There were approximately 371 boys teams and 371 girls teams competing in Washington as of Dec 19, 2005. This would mean 1,855 girl starters and an equal number of boy starters. Visualize (see in your mind’s eye) this data set and again imagine trying to make three statements about this data similar to those made for the Whitehouse, Bullard, and Troup teams. How would you do this?

You need to have the heights of each member of the starting line-ups for each boys’ and girls’ basketball teams in the state.

The process is identical to the one we used above; the only difference is that the set of numbers is much larger. (Using a frequency table would enable us to better handle the large amounts of data.)

______

  1. Describe the difficulties (problems) you might have and some ways you could overcome the difficulties.

Accurately collecting the necessary data (the heights of each member of the starting lineups for each team in the state) might prove problematical. In addition, some teams change their starting line-ups due to their opponent, injuries, or other circumstances.

Keeping track of the large amounts of data might also be difficult. Using a frequency table or a computer spreadsheet might help.


14. It seems reasonable to want to report the mean height of starting high school basketball players in

the state of Washington. Would it be reasonable to report one mean height for high school