Students whose mathematics curriculum has been consistent with the recommendations in Principles and Standards should enter high school having designed simple surveys and experiments, gathered data, and graphed and summarized those data in various ways. They should be familiar with basic measures of center and spread, able to describe the shape of data distributions, and able to draw conclusions about a single sample. Students will have computed the probabilities of simple and some compound events and performed simulations, comparing the results of the simulations to predicted probabilities.
In grades 912 students should gain a deep understanding of the issues entailed in drawing conclusions in light of variability. They will learn more-sophisticated ways to collect and analyze data and draw conclusions from data in order to answer questions or make informed decisions in workplace and everyday situations. They should learn to ask questions that will help them evaluate the quality of surveys, observational studies, and controlled experiments. They can use their expanding repertoire of algebraic functions, especially linear functions, to model and analyze data, with increasing understanding of what it means for a model to fit data well. In addition, students should begin to understand and use correlation in conjunction with residuals and visual displays to analyze associations between two variables. They should become knowledgeable, analytical, thoughtful consumers of the information and data generated by others.
As students analyze data in grades 912, the natural link between statistics and algebra can be developed further. Students' understandings of graphs and functions can also be applied in work with data.
Basic ideas of probability underlie much
of statistical inference. Probability is linked to other topics in high
school mathematics, especially counting techniques (Number and Operations),
area concepts (Geometry), the binomial theorem, and relationships between
functions and the area under their graphs (Algebra). Students should learn
to determine the probability of a sample statistic for a known population
and to draw simple inferences about a population from randomly generated
Students' experiences with surveys and experiments in lower grades should prepare them to consider issues of design. In high school, students should design surveys, observational studies, and experiments that take into consideration questions such as the following: Are the issues and questions clear and unambiguous? What is the population? How should the sample be selected? Is a stratified sample called for? What size should the sample be? Students should understand the concept of bias in surveys and ways to reduce bias, such as using randomization in selecting samples. Similarly, when students design experiments, they should begin to learn how to take into account the nature of the treatments, the selection of the experimental units, and the randomization used to assign treatments to units. Examples of situations students might consider are shown in figure 7.22. »
Nonrandomness in sampling may also limit the conclusions that can be drawn from observational studies. For instance, in the observational study example, it is not certain that the number of people riding trains reflects the number of people who would ride trains if more were available or if scheduling were more convenient. Similarly, it would be inappropriate to draw conclusions about the percentage of the population that ice skates on the basis of observational studies done either » in Florida or in Quebec. Students need to be aware that any conclusions about cause and effect should be made very cautiously in observational studies. They should also know how certain kinds of systematic observations, such as random testing of manufacturing parts taken from an assembly line, can be used for purposes of quality control.
In designed experiments, two or
more experimental treatments (or conditions) are compared. In order for
such comparisons to be valid, other sources of variation must be controlled.
This is not the situation in the tire example, in which the front and
rear tires are subjected to different kinds of wear. Another goal in designed
experiments is to be able to draw conclusions with broad applicability.
For this reason, new tires should be tested on all relevant road conditions.
Consider another designed experiment in which the goal is to test the
effect of a treatment (such as getting a flu shot) on a response (such
as getting the flu) for older people. This is done by comparing the responses
of a treatment group, which gets treatment, with those of a control group,
which does not. Here, the investigators would randomly choose subjects
for their study from the population group to which they want to generalize,
say, all males and females aged 65 or older. They would then randomly
assign these individuals to the control and treatment groups. Note that
interesting issues arise in the choice of subjects (not everyone wants
to or is able to participatecould this introduce bias?) and in the
concept of a control group (are these seniors then at greater risk of
getting the flu?).
Describing center, spread, and shape is essential to the analysis of both univariate and bivariate data. Students should be able to use a variety of summary statistics and graphical displays to analyze these characteristics.
The shape of a distribution of
a single measurement variable can be analyzed using graphical displays
such as histograms, dotplots, stem-and-leaf plots, or box plots. Students
should be able to construct these graphs and select from among them to
assist in understanding the data. They should comment on the overall shape
of the plot and on points that do not fit the general shape. By examining
these characteristics of the plots, students should be better able to
explain differences in measures of center (such as mean or median) and
spread (such as standard deviation or interquartile range). For example,
students should recognize that the statement "the mean score on a test
was 50 percent" may cover several situations, including the following:
all scores are 50 percent; half the scores are 40 percent and half the
scores are 60 percent; half the scores are 0 percent and half the scores
are 100 percent; one score is 100 percent and 50 scores are 49 percent.
Students should also recognize that the sample mean and median can differ
greatly for a skewed distribution. They should understand that for data
that are identified by categoriesfor example, gender, favorite color,
or ethnic originbar graphs, pie charts, and summary tables often
display information about the relative frequency or percent in each category.
Students should learn to apply their knowledge of linear transformations from algebra and geometry to linear transformations of data. They should be able to explain why adding a constant to all observed values in a sample changes the measures of center by that constant but does not » change measures of spread or the general shape of the distribution. They should also understand why multiplying each observed value by the same constant multiplies the mean, median, range, and standard deviation by the same factor (see the related discussion in the "Reasoning and Proof" section of this chapter).
The methods used for representing univariate measurement data also can be adapted to represent bivariate data where one variable is categorical and the other is a continuous measurement. The levels of the categorical variable split the measurement variable into groups. Students can use parallel box plots, back-to-back stem-and-leaf, or same-scale histograms to compare the groups. The following problem from Moore (1990, pp. 1089) illustrates conclusions that can be drawn from such comparisons:
U.S. Department of Agriculture regulations group hot dogs into three types: beef, meat, and poultry. Do these types differ in the number of calories they contain? The three boxplots below display the distribution of calories per hot dog among brands of the three types. The box ends mark the quartiles, the line within the box is the median, and the whiskers extend to the smallest and largest individual observations. We see that beef and meat hot dogs are similar but that poultry hot dogs as a group show considerably fewer calories per hot dog.
Analyses of the relationships between two sets of measurement data are central in high school mathematics. These analyses involve finding functions that "fit" the data well. For instance, students could examine the scatterplot of bivariate measurement data shown in figure 7.23 and consider what type of function (e.g., linear, exponential, quadratic) might be a good model. If the plot of the data seems approximately linear, students should be able to produce lines that fit the data, to compare several such lines, and to discuss what best fit might mean. This analysis includes stepping back and making certain that what is being done makes sense practically.
The dashed vertical line segments in figure 7.23 represent residualsthe differences between the y-values predicted by the linear model and » the actual y-valuesfor three data points. Teachers can help students explore several ways of using residuals to define best fit. For example, a candidate for best-fitting line might be chosen to minimize the sum of the absolute values of residuals; another might minimize the sum of squared residuals. Using dynamic software, students can change the position of candidate lines for best fit and see the effects of those changes on squared residuals. The line illustrated in figure 7.23, which minimizes the sum of the squares of the residuals, is called the least-squares regression line. Using technology, students should be able to compute the equation of the least-squares regression line and the correlation coefficient, r.
Students should understand that
the correlation coefficient r gives information about (1) how
tightly packed the data are about the regression line and (2) about the
strength of the relationship between the two variables. Students should
understand that correlation does not imply a cause-and-effect relationship.
For example, the presence of certain kinds of eye problems and the loss
of sensitivity in people's feet can be related statistically. However,
the correlation may be due to an underlying cause, such as diabetes, for
both symptoms rather than to one symptom's causing the other.
Once students have determined
a model for a data set, they can use the model to make predictions and
recognize and explain the limitations of those predictions. For example,
the regression line depicted in figure 7.23 has the equation y = 0.33x 93.9,
where x represents the number of screens and y represents
box-office revenues (in units of $10 000). To help students understand
the meaning of the regression line, its role in making predictions and
inferences, and its limitations and possible extensions, teachers might
ask questions like the following:
A parameter is
a single number that describes some aspect of an entire population, and
a statistic is an estimate of that value computed from some
sample of the population. To understand terms such as margin of error
in opinion polls, it is necessary to understand how statistics, such
as sample proportions, vary when different random samples are chosen from
a population. Similarly, sample means computed from measurement data vary
according to the random sample chosen, so it is important to understand
the distribution of sample means in order to assess how well a specific
sample mean estimates the population mean.
Understanding how to draw inferences about a population from random samples requires understanding how those samples might be distributed. Such an understanding can be developed with the aid of simulations. Consider the following situation: »
Suppose that 65% of a city's registered voters support Mr. Blake for mayor. How unusual would it be to obtain a random sample of 20 registered voters in which at most 8 support Mr. Blake?
Here the parameter for the population is known: 65 percent of all registered voters support Mr. Blake. The question is, How likely is a random sample with a very different proportion (at most 8 out of 20, or 40%) of supporters? The probability of such a sample can be approximated with a simulation. Figure 7.24 shows the results of drawing 100 random samples of size 20 from a population in which 65 percent support Mr. Blake.
In the situation just described, a parameter of the population was known and the probability of a particular sample characteristic was estimated in order to understand how sampling distributions work. However, in applications of this idea in real situations, the information about a population is unknown and a sample is used to project what that information might be without having to check all the individuals in the population. For example, suppose that the proportion of registered voters supporting Mr. Blake was unknown (a realistic situation) and that a pollster wanted to find out what that proportion might be. If » the pollster surveyed a sample of 20 voters and found that 65 percent of them support the candidate, is it reasonable to expect that about 65 percent of all voters support the candidate? What if the sample was 200 voters? 2000 voters? As indicated above, the proportion of voters who supported Mr. Blake could vary substantially from sample to sample in samples of 20. There is much less variation in samples of 200. By performing simulations with samples of different sizes, students can see that as sample size increases, variation decreases. In this way, they can develop the intuitive underpinnings for understanding confidence intervals.
A similar kind of reasoning about
the relationship between the characteristics of a sample and the population
from which it is drawn lies behind the use of sampling for monitoring
process control and quality in the workplace.
In high school, students can apply the concepts of probability to predict the likelihood of an event by constructing probability distributions for simple sample spaces. Students should be able to describe sample spaces such as the set of possible outcomes when four coins are tossed and the set of possibilities for the sum of the values on the faces that are down when two tetrahedral dice are rolled.
High school students should learn to identify mutually exclusive, joint, and conditional events by drawing on their knowledge of combinations, permutations, and counting to compute the probabilities associated with such events. They can use their understandings to address questions such as those in following series of examples.
The diagram below shows the results of a two-question survey administered to 80 randomly selected students at Highcrest High School.
High school students should learn to compute expected values. They can use their understanding of probability distributions and expected value to decide if the following game is fair:
You pay 5 chips to play a game. You roll two tetrahedral dice with faces numbered 1, 2, 3, and 5, and you win the sum of the values on the faces that are not showing.
Teachers can ask students to discuss whether they think the game is fair and perhaps have the students play the game a number of times to see if there are any trends in the results they obtain. They can then have the students analyze the game. First, students need to delineate the sample space. The outcomes are indicated in figure 7.25. The numbers on the first die are indicated in the top row. The numbers on the second die are indicated in the first column. The sums are given in the interior of the table. Since all outcomes are equally likely, each cell in the table has a probability of 1/16 of occurring.
If a player pays a five-chip fee to play the game, on average, the player will win 0.5 chips. The game is not statistically fair, since the player can expect to win.
Students can also use the sample space to answer conditional probability questions such as "Given that the sum is even, what is the probability that the sum is a 6?" Since ten of the sums in the sample space are even and three of those are 6s, the probability of a 6 given that the sum is even is 3/10.
The following situation, adapted from Coxford et al. (1998, p. 469), could give rise to a very rich classroom discussion of compound events.
In a trial in Sweden, a parking officer testified to having noted the position of the valve stems on the tires on one side of a car. Returning later, the officer noted that the valve stems were still in the same position. The officer noted the position of the valve stems to the nearest "hour." For example, in figure 7.26 the valve stems are at 10:00 and at 3:00. The officer issued a ticket for overtime parking. However, the owner of the car claimed he had moved the car and returned to the same parking place.
The judge who presided over the trial made the assumption that the wheels move independently and the odds of the two valve stems returning to their previous "clock" positions were calculated as 144 to 1. The driver was declared to be innocent because such odds were considered insufficienthad all four valve stems been found to have returned to their previous positions, the driver would have been declared guilty (Zeisel 1968). »
Students could also explore the effect of more-precise measurements on the resulting probabilities. They could calculate the probabilities if, say, instead of recording markings to the nearest hour on the clockface, the markings had been recorded to the nearest half or quarter hour. This line of thinking could raise the issue of continuous distributions and the idea of calculating probabilities involving an interval of values rather than a finite number of values. Some related questions are, How could a practical method of obtaining more-precise measurements be devised? How could a parking officer realistically measure tire-marking positions to the nearest clock half-hour? How could measurement errors be minimized? These could begin a discussion of operational definitions and measurement processes.
Students should be able to investigate the following question by using a simulation to obtain an approximate answer:
How likely is it that at most 25 of the 50 people receiving a promotion are women when all the people in the applicant pool from which the promotions are made are well qualified and 65% of the applicant pool is female?
Those students who pursue the study of probability will be able to find an exact solution by using the binomial distribution. Either way, students are likely to find the result rather surprising.
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