Standards - Mathematics

2019 - Mathematics Course of Study Document
Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.17

Use the Statistical Problem Solving Cycle to answer real-world questions.

Unpacked Content

Knowledge

Students know:

  • Techniques for producing a sampling distribution.
  • Properties of a normal distribution.

Skills

Students are able to:

  • Produce a sampling distribution.
  • Reach accurate conclusions regarding the population from the sampling distribution.
  • Accurately create and interpret a confidence interval based on observations from the sampling distribution.

Understanding

Students understand that:

  • The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable’s distribution in the population.
  • A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.

Vocabulary

  • Bootstrapping
  • Population mean
  • Approximately normal
  • Standard deviation
  • Confidence interval
Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.18

Construct a probability distribution based on empirical observations of a variable.

Example: Record the number of student absences in class each day and find the probability that each number of students will be absent on any future day.

Unpacked Content

Knowledge

Students know:

  • Techniques for conducting randomized experiments.
  • Techniques for conducting simulations of randomized experiment situations.

Skills

Students are able to:

  • Design and conduct randomized experiments with two treatments.
  • Draw conclusions from comparisons of the data of the randomized experiment.
  • Design, conduct, and use the results from simulations of a randomized experiment situation to evaluate the significance of the identified differences.

Understanding

Students understand that:

  • Differences of two treatments can be justified by a significant difference of parameters from a randomized experiment.
  • Statistical analysis and data displays often reveal patterns in data or populations, enabling predictions.

Vocabulary

  • Randomized experiment
  • Significant
  • Parameters
Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.18c

Find the mean, standard deviation, median, and interquartile range of a probability distribution and make long-term predictions about future possibilities. Determine which measures are most appropriate based upon the shape of the distribution.

Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.19

Construct a sampling distribution for a random event or random sample.

Examples: How many times do we expect a fair coin to come up heads“ in 100 flips and on average how far away from this expected value do we expect to be on a specific set of flips? What do we expect to be the average height for a random sample of students in a local high school given the mean and standard deviation of the heights of all students in the high school?“

Unpacked Content

Knowledge

Students know:

  • The circumference of any circle is 2πr and therefore, the circumference of a unit circle is 2π.

Skills

Students are able to:

  • Translate between arc length and central angle measures in circles.

Understanding

Students understand that:

  • Radians measure angles as a ratio of the arc length to the radius.
  • The unit circle has a circumference of 2π which aids in sense making for angle measure as revolutions (one whole revolution measures 2π radians) regardless of radius.
  • Use of the unit circle gives a one-to-one ratio between arc length and the measure of the central angle, putting the angle in direct proportion to the arc length, and that the circle can then be divided up to find the radian measure of other angles.

Vocabulary

  • Radian measure
  • Constant of proportionality
  • Unit circle
  • Intercepted arc
Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.20

Perform inference procedures based on the results of samples and experiments.

Unpacked Content

Knowledge

Students know:
  • how to calculate the margin of error in a statistical sample.
  • how to express the confidence interval for a statistical sample or experiment.
  • how to perform a significance test.
  • how to use the results of the significance test to either support or refute a claim.

Skills

Students are able to:
  • Calculate the margin of error.
  • Determine a confidence interval.
  • perform a significance test.
  • Use the results of the significance test to support or refute the null hypothesis.

Understanding

Students understand that:
  • hypothesis testing is used to evaluate claims about a population.
    • a confidence interval helps to determine the size of a sample needed to provide accurate calculations.

Vocabulary

  • Point Estimator
  • Margin of Error
  • Confidence Interval
  • Significance Test
  • Null Hypothesis
  • Hypothesis
Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.20d

Interpret the significance level of a test in the context of error probabilities, and use the results to make strategic decisions.

Example: How do you reduce the rate of human error on the floor of a manufacturing plant?

Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.21

Critique the validity of reported conclusions from statistical studies in terms of bias and random error probabilities.

Unpacked Content

Knowledge

Students know:
  • what constitutes a bias in a statistical study.
  • The accepted statistical process that can be used to analyze results from a statistical study.

Skills

Students are able to:
  • Calculate and interpret results from a statistical study.
  • Calculate random error probability.
  • Identify biases that can affect the validity of a mathematical argument.

Understanding

Students understand that:
  • A valid mathematical argument is based on rigorous statistical processes.
  • bias and random error probability can affect the validity of a mathematical argument.

Vocabulary

  • Validity
  • Bias
  • Random Error Probability
Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.22

Conduct a randomized study on a topic of student interest (sample or experiment) and draw conclusions based upon the results.

Example: Record the heights of thirty randomly selected students at your high school. Construct a confidence interval to estimate the true average height of students at your high school. Question whether or not this data provides significant evidence that your school’s average height is higher than the known national average, and discuss error probabilities.

Unpacked Content

Knowledge

Students know:
  • how to design a statistical study.
  • how to collect data.
  • how to construct confidence intervals.
  • how to use confidence intervals to make decisions.
  • how to conduct a hypothesis test.
  • how to use the hypothesis test to make a decision.
  • how to communicate the results of their study.

Skills

Students are able to:
  • Collect data appropriately.
  • Calculate and interpret results from a statistical study.
  • Calculate margin of error.
  • Construct confidence intervals.
  • Conduct hypothesis tests.

Understanding

Students understand:
  • how to design and conduct a statistical study and are able to communicate their findings to others.

Vocabulary

  • Randomized Study
  • Sample
  • Experiment

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