Chapter 15 9-12

New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

STANDARD 15: PROBABILITY AND STATISTICS

All students will develop their understanding of probability and statistics through experiences which enable them to systematically collect, organize, and describe sets of data, to use probability to model situations involving random events, and to make inferences and arguments based on analysis of data and mathematical probabilities.

9 - 12 Overview

Probability and statistics hold the key for enabling our students to understand, process, and interpret the vast amounts of quantitative data that exist all around them. To be able to judge the truth of a data-supported argument presented to them, to discern the believability of a persuasive advertisement that talks about the results of a survey of all of the users of a particular product, or to be knowledgeable consumers of the data-intensive government and electoral statistics that are ever-present, students need the skills that they can learn in a well-conceived probability and statistics curriculum strand.

The key components of statistics applicable here are recognizing the significance of sampling in assessing the validity of statistical claims, using measures of dispersion and correlation in analyzing data, using curve fitting to interpolate and predict from data, and when presented with a problem, designing a statistical experiment, conducting it, interpreting the data and communicating the outcomes. The key components of probability at this level are using relative frequency and probability to solve problems, using simulations to estimate probabilities, understanding the use of probability distributions in real-world situations, and understanding and applying the law of large numbers.

The field of statistics is relatively new. Beyond the work of scientists, there was little use for gathering and analyzing data. Monarchs cared little what their subjects preferred and shops carried the basics for existence. As governments moved to democratic forms which responded to the needs and desires of the people and officials required the support of an electorate, it became more important to gather and analyze data regarding such issues. As the United States became more affluent and industry became more efficient, companies began competing for the consumer's dollar. Advertisements began making claims about the quality of their products quoting figures which supported them. As advertising companies recognized the significance of television, the need grew to gather data and report it in such a way that people believed the quality of their product was excellent and that they had to have it to achieve the status they desired. While an understanding of statistics is still needed to deal with all these issues, it has permeated almost every level of our lives. To be a successful member of this information age, students need to leave high school with an understanding of probability and statistics equal to that most people received previously only at the college level.

By the time students enter high school, they should have mastered basic descriptive statistical methods. In earlier grades, they used statistics in such a variety of areas that they could set up a study, gather the data, and appropriately analyze and report their findings. Throughout grades 9 to 12, students should have numerous opportunities to continue to practice these skills in a variety of ways, but they need to be extended as well. The best way to learn statistics is to be involved. The content should be developed through a problem-centered approach. For example, if students are required to generate a report on two sets of data which have the same measures of central tendency only to find later they have very different dispersion, they recognize the need for some way to identify that difference. As students learn new algebraic functions, they might revisit a problem they previously had modeled linearly and apply a different model. For example, they may have linearly modeled the series of winning times of the men's Olympic marathon but now understand that there would probably be a limiting time and so attempt to fit a quadratic or logarithmic curve instead.

Today's society is full of opportunities for people to take risks. The focus of most news reporting is all the horror and gore existing that day. John Allen Paulos, in his book, Innumeracy, cites numerous problems associated with a lack of understanding of probability. If people are to make appropriate decisions, then they must understand the relationship of probability to real situations and be able to weigh the consequences against the odds. As with statistics, probability needs to be experienced, not memorized. Work done at this level should provide insight into the use of probability and probability distributions in a variety of real-world situations. The normal curve presents interesting opportunities to examine uses and abuses of mathematics, especially in their academic lives.

The work students do in both probability and statistics must be centered around problems for the meaning is most clear through the contexts of problems. They must have access to appropriate technology and the ability to use it. The technology must not be limited to number crunching and analysis of data but must provide for the acquisition of appropriate data. Initial thoughts would be access to Internet and other on-line databases or CD resources such as almanacs, but computers which simulate experiments not possible in a high school lab or calculator/computer data gatherers such as Texas Instrument's CBL are needed to provide maximum opportunities. Whenever possible, real data gathered from school, the community, or cooperating businesses should be used.

Probability and statistics provide a rich opportunity to integrate with other mathematics content and other disciplines. This content provides the opportunity to generate the numbers and situations which should be used in other areas such as geometry, algebra, and functions. The goal to have students become effective members of a democratic society requires them to practice and participate in decision-making experiences. The ability to make intelligent decisions rests on an understanding of statistics and probability. Having students use this content in their experiences in social studies, science, and other disciplines should be supported with an expectation that it also is used in their social and school interactions.

The topics that should comprise the probability and statistics focus of the mathematics program in grades 9 through 12 are:

designing, conducting, and interpreting statistical work to solve problems

analyzing data using range, measures of central tendency, and dispersion

applying probability dispersions in real situations

evaluate arguments based upon their knowledge of sampling and data analysis

interpolate and/or extrapolate from data using curve fitting

using simulations to estimate probabilities

using the law of large numbers

STANDARD 15: PROBABILITY AND STATISTICS

9 - 12 Expectations and Activities

The expectations for these grade levels appear below in boldface type. Each expectation is followed by activities which illustrate how the expectation can be addressed in the classroom.

Building upon K-8 expectations, experiences in grades 9-12 will be such that all students:

S. estimate probabilities and predict outcomes from real-world data.

Students are asked to compute the probability that a given lottery ticket for a senior class raffle to raise money for the senior trip will be the winning ticket. The class will be printing 500 tickets that they will sell for $1 each. First prize is a stereo worth $150. Second prize is a $100 shopping spree in the local gap store. Third prize is a $50 gift certificate to The Golden Goose restaurant.
Students determine the area of an irregular closed figure drawn on a large sheet of paper using the Monte Carlo method. Each person in the group drops a handful of pennies over their shoulder (without looking) on to the paper containing the figure. They count the number of coins on the paper (total shots) and the number within the figure (hits). They thus produce the ratio of hits to total shots and multiply the fraction by the area of the paper to estimate the area of the figure.

T. understand sampling and recognize its role in statistical claims.

While studying United States history, students read about the prediction in the 1936 presidential race that Alfred Landon would defeat incumbent President Franklin Roosevelt. They raise questions as to why that prediction was so far off and how TV stations can forecast winners of some elections with a very small percentage of the voting results reported. Students contact local radio and TV stations and newspapers to discover how they determine their sample population.
After reading the chapter on sampling in the book How to Lie With Statistics by Darryl Huff, students bring in ads, graphs, and charts from newspapers, and articles which all make statements or claims allegedly based on data. Students examine the articles for information about the sample and identify those claims which have little or no substantiation. They go further and discuss how the sample populations chosen could have influenced the outcomes.

U. understand and apply measures of dispersion and correlation.

Students are presented with data gathered by an archaeologist at several sites. The data identifies the number of flintstones found at each site and the number of charred bones. The archaeologist claimed that the data showed that the flintstones were used to light the fires that charred the bones. Students produce a scatterplot, find the correlation between the two sets of figures, and use their work to support or criticize the claim.
In their journals, students respond to the claim that children with bigger feet spell better. They discuss whether they believe the claim is true, how statistics might have led to this claim, and whether it has any importance to a philosophy of language teaching.

V. design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes.

Out of a discussion among some members of the class, a question as to which are the most popular cars in the community arises. The students work in cooperative groups to design an experiment that will gather the data, analyze the data, and design an appropriate report format for their results.
Intrigued by the question How long would it take dominoes set up one inch apart across the room to fall?, the class designs an experiment to gather data on smaller sets of dominoes and then extrapolates to estimate the answer.
Students had just finished a unit in which they discussed the capture-recapture method for estimating the populations of wildlife. Part of their assessment for the unit is a project where they work in groups to design and conduct a simulation of the capture-recapture method. One group uses the method to determine the number of lollipops in a large bag.

W. use curve fitting to interpolate and predict from data.

Students are presented with this data comparing a student's test grade to the number of hours studied.
```
     Hours     1    2    3    4    4    6    8    9    10    10    12    12
     Grade     60   55   65   65   77   80   83   80   75    90    72    80
```
Earlier in the year they had produced a line of best fit for the data, but they had recognized that it was not a good model for the data. Now the students fit a quadratic curve to the data and discuss the advantages it has over the straight line.
Students conduct an experiment where they suspend a weight on a string from a hook in the doorway. They swing their homemade pendulum and time how long it takes for it to swing 10 times. They had performed the experiment in 8th grade and used the median-median line fit method to model the data. In this revisitation of the problem, the teacher insists they use very short lengths and very long lengths in addition to the various ones in the middle. When the data is graphed, it becomes apparent that the data is not linear and would fit a quadratic curve better.
Students perform the Introductory Physical Science experiment where water is cooled by adding ice cubes and stirring. The TI Calculator Based Lab temperature probe is attached to the TI-82 which is programmed to gather the data. Students (or each group of students) link their calculators to the original one to transfer the data to their calculators. They use the stat calc functions to perform a quadratic fit, an exponential fit, and a logarithmic fit and use the function graph capabilities to determine which is the best model.

X. use relative frequency and probability, as appropriate, to represent and solve problems involving uncertainty.

fter a unit where dependent and independent events were detailed, students are challenged by a problem containing this excerpt from The Miami Herald of May 5, 1983.

An airline jet carrying 172 people between Miami and Nassau lost its engine oil, power, and 12,000 feet of altitude over the Atlantic Ocean before a safe recovery was made.
When all three engines' low oil pressure warning lights all lit up at nearly the same time, the crew's initial reaction was that something was wrong with the indicator system, not the oil pressure.
They considered the possibility of a malfunction in the indication system because it's such an unusual thing to see all three with low pressure indications. The odds are so great that you won't get three indications like this. The odds are way out of sight, so the first thing you would suspect is a problem with the indication system.

Aviation records show that the probability of an engine failure in any particular hour is about 0.00004. If the failures of three engines were independent, what would the be the probability of them failing within one hour? Discuss why the speaker in the article would refer to such a probability as "way out of sight." Discuss situations which might make the failures of three engines not independent events.
Students keep a record of their trips through the town and whether they get stopped at each of the four traffic lights. After one month, the data is grouped and studied. They use their data to determine whether the timing of the lights is independent or not.
While discussing the issue of mandatory drug testing in social studies, students examine the probability of misdiagnosing people as having AIDS with a test that would identify 99% of those who are true positive and misdiagnose 3% of those who don't have AIDS. They examine situations where the prevalence of the disease is 50%, 10%, and 1% using 100,000 people as a base. They discuss the fact that, at the 1% level, 75% of the people identified as having AIDS would be false positives, the implications that has on mandatory testing, and potential ways to improve the predictive value of testing.

Y. use simulations to estimate probabilities.

Students write a computer program to simulate winning the New Jersey Pick 6 lottery. The students enter the winning numbers and the computer generates sets of 6 numbers until it hits the winning combination. The computer prints out the number of sets generated including the winning one. Students run the program several times, attempting to verify the theoretical probability they derived.
In the process of solving the hard disk problem mentioned in AA below, students write a computer program which simulates the situation 1000 times. For each trial, they have the program generate a random number for each of the 36 disks in the box. Numbers in the range 0 < n =< .3 were taken as bad disks and the remainder were good disks. The program computes the average life span from the trials and stores it in an array. After 1000 trials, it prints a histogram of the data.

Z. create and interpret discrete and continuous probability distributions and understand their application to real-world situations.

Students work on a project to pick one form of insurance (life, car, home). They are to determine the variables which affect the premiums they would need to pay for this type of insurance and what it would cost for them to obtain it. From their research, they are to write an essay summarizing how insurance companies use statistics and probabilities to determine their rates.
An article in Consumer Reports indicates that 25% of 5-lb bags of sugar from a particular company are underweight. The class works with the local supermarket to develop and perform a consumer research project. Each group is given a commodity (potato chips, 5 lb bags of sugar) to study. They design a method for randomly selecting and testing whether the product matches the claimed specifications or not. They use their data to determine the probability that a randomly selected bag would be underweight.

AA. describe the normal curve in general terms and use its properties to answer questions about sets of data that are assumed to be normally distributed.

Students represent a typical student in the school. Students first select a random sample of 30 students in their school. They survey their sample for information they believe necessary to identify what would be "typical". They use appropriate displays and descriptive statistics to support their representation of a typical student.
Students are given the administrator's summary of the school's standardized tests. Each group is given one area on which they are to focus. They prepare a presentation they would give to the Board of Education discussing the comparisons between local norms, national norms, suburban norms, and independent norms using their understanding of normal distribution, percentile ranks, and graphical displays.
Students are introduced to the central limit theorem through this problem:

A worker on the assembly line at Western Digital is involved in industrial sabotage by weakening a soldering joint that causes a hard drive to fail after 5 hours of use. At his station, he actually comes in contact with 30% of the drives produced. The other 70% will last 100 hours. If they are packed randomly in boxes of 36, what would be the average expected lifespan of the drives in the box?

Students prepare simulations of the problem and discover that the answers fall into a normal curve with a mean of approximately 70 hours.

BB. make predictions based on extrapolation and interpolation of data.

n ornithologist in the local bird sanctuary feeds a special nutrient to 26 of the 257 pelicans in the bird sanctuary and tags those that to whom she feeds it. A week later she captures a pelican in the sanctuary. Students determine the probability that the bird is one which was tagged. They further discuss the probability that a second one captured would be tagged considering the different possibilities for the first one and whether she releases the first one or not.
Students use the demographics of their community from the most recent census, the current number of funeral directors, and a mortality table to determine whether a new funeral home would find it profitable to open there.

CC. understand and use the law of large numbers.

Students are given two dice, each a different color. They keep track of the result for each of them as well as the total after each roll. After a large number of rolls they compare their relative frequencies to the expected outcomes for a larger number of choices. They use their comparisons to determine whether their dice can be considered "fair."
Students are presented with a paper containing a gambler's formula (When playing roulette, bet red. If red does not win, double the bet on red. Continue in this manner.). They evaluate whether the formula makes sense, identify potential problems, and limitations, and discuss the fallacy that the odds improve for red to appear on the next roll every time red doesn't win.