Montana Math MT.PLUS.DATA.INF.2.b

MathGrades 9–12Statistical Inference Using Simulation

The Standard

MT.PLUS.DATA.INF.2
Estimate a population parameter from a representative sample by: Understanding why the sample statistic is the best estimate for the associated population parameter, Understanding that sampling variability introduces uncertainty in the estimate, and account for the uncertainty with a confidence interval by:  Using resampling with replacement from an observed sample to produce a sampling distribution,  Verifying that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough,  Verifying that 95% of sample means are within two standard deviations of the sampling distribution from the population mean, and  Creating and interpreting a 95% confidence interval based on an observed mean from a sampling distribution.
Estimate a population parameter from a representative sample by: Understanding that sampling variability introduces uncertainty in the estimate, and account for the uncertainty with a confidence interval by:  Using resampling with replacement from an observed sample to produce a sampling distribution,  Verifying that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough,  Verifying that 95% of sample means are within two standard deviations of the sampling distribution from the population mean, and  Creating and interpreting a 95% confidence interval based on an observed mean from a sampling distribution.

2026 Montana Mathematics Content Standards · Data Reasoning

Teacher's field guide

What This Standard Means

What Students Need to Do

Students use a random, representative sample to estimate a population average. They resample from their observed sample with replacement and graph the resample means. They use the spread of those means to make a 95% confidence interval.

What Mastery Looks Like

In a simulation with a known population mean, students show that repeated sample means cluster near it. With a large enough sample, they check that the means are roughly bell-shaped and about 95% fall within two standard deviations of the population mean. They use the observed mean plus or minus twice the bootstrap standard deviation to estimate the population mean.

Common Misconceptions

Students may draw bootstrap resamples without replacement or expect their means to center on the unknown population mean. They may think the interval contains 95% of individual data values. Some say a calculated interval has a 95% chance of containing the fixed population mean.

How to Assess It

Exit ticket: A random sample of 40 commute times has a mean of 24 minutes. Five hundred bootstrap resamples give a roughly bell-shaped set of means with a standard deviation of 2 minutes. Ask students to calculate a 95% confidence interval, explain what it estimates, and say why the resample means center near 24.

Lesson moves

Ways to Teach It

  1. Give pairs a 100-value spreadsheet; graph means from 100 random samples of 30, then bootstrap one sample and compare centers.

  2. Ask, "Why do repeated random samples center near the population mean, while bootstrap samples center near our sample mean?"

  3. Give pairs bootstrap histograms marked with a mean and standard deviation; race to mark two standard deviations on each side and write the interval.

  4. Give students a list of 100 grocery prices; have them sample 30 and estimate the list's mean price with a bootstrap interval.

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