Math 3 Unit 1: Statistics



Approximate Time Frame: 2 – 3 Weeks

Connections to Previous Learning:

Students will build on their understanding of data distributions to see how the normal distribution uses area to estimate frequencies (expressed as probabilities). Students will use different ways of collecting, representing, and displaying data to make comparisons.

Focus of this Unit:

Students will understand how visual displays and summary statistics relate to different types of data and to probability distributions. They will understand methods of collecting data (including sample surveys, experiments, and simulations), organizing, summarizing, analyzing, and presenting data. They will also understand how randomness and design affect conclusions.

From the High School, Statistics and Probability Progression Document pp. 4-7:

Data on heights of adults are available for anyone to look up. But how can we answer questions about standardized test scores when individual of scores is given? Students should now realize that we can do this only because such standardized scores generally have a distribution that is mound-shaped and somewhat symmetric, i.e., approximately normal. For example, SAT math scores for a recent about 16% of the scores are above 632. In fact, students should be aware that technology now allows easy computation of any area under normal curve. “If Alicia scored 680 on this SAT mathematics exam, what proportion of students taking the exam scored less than she scored?” (Answer: about 92%)

Summarize, represent, and interpret data on two categorical and quantitative variables. As with univariate data analysis, students now take a deeper look at bivariate data, using their knowledge of proportions to describe categorical associations and using their knowledge of functions to fit models to quantitative data. The table below shows statistics from the Center for Disease Control relating HIV risk to age groups. Students should be able to explain the meaning of a row or column total (marginal), a row or column percentage (conditional) or a “total” percentage (joint). They should realize that possible associations between age and HIV risk are best explained in terms of the row or column conditional percentages. Are the comparisons of percentages valid when the first age category is much smaller (in years) than the others?

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Students have seen scatter plots in Grade 8 and now extend that knowledge to fit mathematical models that capture key elements of the relationship between two variables and to explain what the model tells us about that relationship. Some of the data should come from science, as in the examples about cricket chirps and temperature, and tree growth and age, and some from other aspects of their everyday life, e.g., cost of pizza and calories per slice (p. 6).

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If you have a keen ear and some crickets, can the cricket chirps help you predict the temperature? The margin shows data modeled in a scientific investigation of that phenomenon. In this situation, the variables have been identified as chirps per second and temperature in degrees Fahrenheit. The cloud of points in the scatter plot is essentially linear with a moderately strong positive relationship. It looks like there must be something other than random behavior in this association. A model has been formulated: The least squares regression line has been fit by technology. The model is used to draw conclusions: The line estimates that, on average, each added regression line.” chirp predicts an increase of about 3.29 degrees Fahrenheit. But, students must learn to take a careful look at scatter plots, as sometimes the “obvious” pattern does not tell the whole story, and can even be misleading. The margin shows the median heights of growing boys through the ages 2 to 14. The line (least squares regression line) with slope 2.47 inches per year of growth looks to be a perfect fit. S-ID.6c But, the residuals, the collection of differences between the corresponding coordinate on the least squares line and the actual data value for each age, reveal additional information. A plot of the residuals shows that growth does not proceed at a constant rate over that years. What would be a better description of the growth pattern?

It is readily apparent to students, after a little experience with plotting bivariate data, that not all the world is linear. The figure below shows the diameters (in inches) of growing oak trees at various ages (in years). A careful look at the scatter plot reveals some curvature in the pattern, which is more obvious in the residual plot, because the older and larger trees add to the diameter more slowly. Perhaps a curved model, such as a quadratic, will fit the data better than a line. The figure below shows that to be the case.

Would it be wise to extrapolate the quadratic model to 50-yearold trees? Perhaps a better (and simpler) model can be found by thinking in terms of cross-sectional area, rather than diameter, as the measure that might grow linearly with age. Area is proportional to the square of the diameter, and the plot of diameter squared versus age in the margin does show remarkable linearity, but there is always the possibility of a closer fit that students familiar with cube root, exponential, and logarithmic functions could investigate. Students should be encouraged to think about the relationship between statistical models and the real world, and how knowledge of the context to building good models.

Connections to Subsequent Learning: Students will use probabilities to make fair decisions and analyze those decisions using probability concepts.

From the High School Statistics and Probability Progression Document, p. 20:

Careers A few examples of careers that draw on the knowledge discussed in this Progression are actuary, manufacturing technician, industrial engineer or statistician, industrial engineer and production manager. The level of education required for these careers and sources of further information and examples of workplace tasks are summarized in the table below. Information about careers for statisticians in health and medicine, business and industry, and government appears on the web site of the American Statistical Association (careers/index.cfm).

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College Most college majors in the sciences (including health sciences), social sciences, biological sciences (including agriculture), business, and engineering require some knowledge of statistics. Typically, this exposure begins with a non-calculus-based introductory course that would expand the empirical view of statistical inference found in this high school progression to a more general view based on mathematical formulations of inference procedures. (The Advanced Placement Statistics course is at this level.) After that general introduction, those in more applied areas would take courses in statistical modeling (regression analysis) and the design and analysis of experiments and/or sample surveys. Those heading to degrees in mathematics, statistics, economics, and more mathematical areas of engineering would study the mathematical theory of statistics and probability at a deeper level, perhaps along with more specialized courses in, say, time series analysis or categorical data analysis. Whatever their future holds, most students will encounter data in their chosen field—and lots of it. So, gaining some knowledge of both applied and theoretical statistics, along with basic skills in computing, will be a most valuable asset indeed!

|Desired Outcomes |

|Standard(s): |

|Understand and evaluate random processes underlying statistical experiments. |

|S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. |

|S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g. using simulations. For example a model says a spinning coin falls heads up with probability 0.5. Would a result |

|of 5 tails in a row cause you to question the model? |

|Make inferences and justify conclusions from sample surveys, experiments, and observational studies. |

|S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observations studies; explain how randomization relates to each. |

|S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. |

|S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if difference between parameters are significant. |

|S.IC.6 Evaluate reports based on data. |

|Summarize, represent, and interpret data on two categorical and quantitative variables. |

|S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. |

|Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given function or choose a function suggested by the context. Emphasize linear, quadratic, and exponential |

|models. |

|Informally assess the fit of a function by plotting and analyzing residuals. |

|Fit a linear function for a scatter plot that suggests a linear association. |

|Summarize, represent, and interpret data on a single count or measurement variable. |

|S.ID.4 Use the mean and standard deviation of a data set to fit to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use |

|calculators, spreadsheets, and tables to estimate areas under the normal curve. |

|WIDA Standard: (English Language Learners) |

|English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. |

|English language learners benefit from: |

|explicit vocabulary instruction with regard to the components of data representations and processes for collecting, organizing and interpreting data. |

|guidance to connect visual, algebraic and real-world representations to the language surrounding each. |

|Understandings: Students will understand … |

|Statistics is a process of making inferences. |

|Results from a model may or may not be consistent with a real-life simulation of the process. |

|Different data collection methods are appropriate for different situations and randomization relates to each. |

|Data from a sample survey is used to estimate a population mean. |

|Simulations are used to decide if differences between parameters are significant. |

|A scatter plot may be used to represent data with two quantitative variables and determine how the variables are related. |

|The mean and standard deviation of a data set is used to fit a normal distribution. |

|Essential Questions: |

|How is statistics used? |

|When is it appropriate to question the results from a model compared to a real-life simulation? |

|Which data collection method is best used for a specific context? |

|How does randomization relate to a data collection method? |

|How is a population mean estimated from data from a sample survey? |

|When is the difference between parameters significant? |

|From a scatterplot, how are two quantitative variables related? |

|How is a data set fit to a normal curve? |

|Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) |

|1. Make sense of problems and persevere in solving them. |

|*2. Reason abstractly and quantitatively. Students will assign variables to data sets in order to make predictions or inferences. |

|*3. Construct viable arguments and critique the reasoning of others. Students will use a variety of statistical tools to construct and defend logical arguments based on data. |

|*4. Model with mathematics. Students will create visual, tabular and algebraic models to analyze probability and statistical predictions. |

|*5. Use appropriate tools strategically. Technology will be used to estimate areas under the normal curve. |

|6. Attend to precision. |

|7. Look for and make use of structure. |

|8. Look for and express regularity in repeated reasoning. Students will observe regular patterns in distributions of sample statistics. |

| |

|Prerequisite Skills/Concepts: |Advanced Skills/Concepts: |

|Students should already be able to: |Some students may be ready to: |

|Create visual displays of data sets. |Further their exploration of statistical modeling, focusing on the design and analysis of experiments and/or sample |

|Determine probabilities. |surveys. |

|Analyze data using statistical measures. | |

|Knowledge: Students will know… |Skills: Students will be able to … |

|Normal distributions are only appropriate for some data. |Use the mean and standard deviation for a data set to fit the data to a normal curve. |

|Scatter plots can only be used to represent quantitative variables. |Estimate the area under a normal curve using technology and explain the significance of this value in terms of |

|The role of randomization in sample surveys, experiments, and observational studies. |probability and the original context. |

|The difference between variables as quantitative or categorical. |Sketch the function of best fit on a scatter plot and find the function using technology when necessary. |

| |Choose a probability model for a problem context. |

| |Conduct a simulation of a model and determine which results are typical or considered outliers. |

| |Calculate a sample mean or population. |

| |Determine how often the true population mean or proportion is within the margin of error of each sample mean or |

| |proportion. |

| |Conduct a simulation for each group using sample results as the parameters for the distributions. |

|Academic Vocabulary: |

| | | | |

|Critical Terms: | |Supplemental Terms: | |

|Inference | |Population | |

|Population parameter | |Simulation | |

|Random sample | |Model | |

|Statistics | |Event | |

|Randomization | |Experimental probability | |

|Population mean | |Theoretical probability | |

|Sample mean | |Sample survey | |

|Margin of error | |Experiment | |

|Confidence interval | |Histogram | |

|Standard deviation | | | |

|Bias | | | |

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