Pacing Guide Secondary III: Block 1 Sampling and Inference
Recommended Time Frame: 7 weeks
Start Date: August 21, 2013
Estimated End Date: October 11, 2013 Actual End Date:
Background/Helpful Information: The central idea in this block is the idea of inference—how reliable, predictive information can be inferred
about a population from the data from a representative sample of that population. In the fullness of time, students will begin to encounter this idea
in Grade 7 of the Utah Core, but this year’s transition group will not have had this experience. This block will need to build an understanding of
randomness and its role in selecting a “representative” sample from a target population. Given the inherent variability in any population with
respect to a given characteristic of interest (i.e. parameter), eliminating bias in the process of sampling is critical to being able to use statistical
methods to make inferences from data collected.
The block has three parts:
Understanding Randomness and Sampling. Students discover what “random” really means, what kind(s) of variability are associated with
randomness, and that randomness can’t be faked. They use simulations to explore sampling in various contexts.
Assuring Random Sampling in Various Kinds of Data Collection. Students learn about the four types of data collection structures
(simulations, which they’ve already encountered, and sample surveys, observational studies, and experiments) and what needs to be done in each
case to assure that random sampling occurs and that bias (intentional or unintentional) doesn’t infect the sampling process.
Recognizing When Variability of a Sample is (Probably) Not Random. Ultimately, research and data collection efforts are aimed at
discovering characteristics, causes, and influences that aren’t just the result of random variation. We need to have a systematic, mathematicallyreliable means of recognizing when a data sample is “different enough” from what we expect a completely random sample to be so that we can infer
that something statistically significant is probably accounting for the difference. Different data collection structures will lead to differently shaped
distributions due to random variation. Although, students will have informally seen other distributions, particularly the uniform and geometric
distributions (without learning their names), we will focus on the normal distribution in detail as a means of learning how to recognize variability
that is probably not due to randomness alone. Students should understand how the standard deviation is computed and how to do it using
technology. Connections could be made to the least-squares approach for fitting a line to data and the idea of standard deviation as characterizing
the variability of a sample. Students will learn under what conditions is the normal distribution appropriate to assume for a population (that data is
unimodal and relatively symmetric). An analogy to linear regression is apt here as well: Not all bivariate data are best modeled with a line of best
fit, not all data collections for a population are best modeled with a normal distribution. Just as with bivariate data, students must learn to be alert
for signs that the normal distribution may not apply. They will learn the 98-95-99.7 Rule for the normal distribution. They will learn how to figure
out a sample’s z-score (the number of standard deviations it is from the mean) and use it to decide whether or not it is likely to be the result of
something other than random chance. Note that we are not, for non-honors classes, computing probabilities formally, nor applying statistical
hypothesis testing (such a t-tests). Teachers may wish to introduce a z-table of probabilities to aid in conceptual development, but not assess fluency
using such tables to compute probabilities at the non-honors level. Rather, focus on having students visually place a sample mean on a histogram of
a normal population distribution in order to consider how far out into the “tail” of the distribution does it need to be to be reasonably sure that it is
influenced by something more than random variation.
CURRICULUM
INSTRUCTION
ASSESSMENT
(connected background standards from prior years)
7.SP.1: Understand that statistics can be used to gain information
about a population by examining a sample of the population;
generalizations about a population from a sample are valid only if
the sample is representative of that population. Understand that
random sampling tends to produce representative samples and
School Advisory Panel:
http://www.illustrativemathematics.org/illustrations/186
Strict Parents:
http://www.illustrativemathematics.org/illustrations/122
Summative Assessment (s):
Block 1 Growth Assessment Preand Post
Formative Assessment(s):
support valid inferences.
7.SP.2: Use data from a random sample to draw inferences about a
population with an unknown characteristic of interest. Generate
multiple samples (or simulated samples) of the same size to gauge
the variation in estimates or predictions. For example, estimate
the mean word length in a book by randomly sampling words
from the book; predict the winner of a school election based on
randomly sampled survey data. Gauge how far off the estimate or
prediction might be.
Understand and evaluate random processes
underlying statistical experiments.
S.IC.1: Understand that statistics allows inferences to
be made 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 simulation. 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? [Include comparing theoretical and
empirical results to evaluate the effectiveness of a
treatment.]
Make inferences and justify conclusions from
sample surveys, experiments, and
observational studies.
[In earlier grades, students are introduced to different ways of
collecting data and use graphical displays and summary statistics to
make comparisons. These ideas are revisited with a focus on how
the way in which data is collected determines the scope and nature
of the conclusions that can be drawn from that data. The concept of
statistical significance is developed informally through simulation
as meaning a result that is unlikely to have occurred solely as a
result of random selection in sampling or random assignment in an
experiment.]
S.IC.3: Recognize the purposes of and differences
among sample surveys, experiments, and observational
studies; explain how randomization relates to each.
S.IC.4: Use data from a sample survey to estimate a
Why Randomize:
http://www.illustrativemathematics.org/illustrations/191
Do you Fit in This Car:
http://www.illustrativemathematics.org/illustrations/1020
Should We Send Out a Certificate:
http://www.illustrativemathematics.org/illustrations/1218
Birthday Paradox
Does Your iPod Play Favorites?
Collect All
Panda Population
What is Random Behavior?
Homework, Checkpoints,
Quizzes, Tests
Performance
Assessment(s):
population mean or proportion; develop a margin of
error through the use of simulation models for random
sampling. [Focus on the variability of results from experiments—
that is, focus on statistics as a way of dealing with, not eliminating,
inherent randomness.]
S.IC.5: Use data from a randomized experiment to
compare two treatments; use simulations to decide if
differences between parameters are significant.
S.IC.6: Evaluate reports based on data.
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 it 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. [While students may have heard of
the normal distribution, it is unlikely that they will have prior
experience using it to make specific estimates. Build on students’
understanding of data distributions to help them see how the
normal distribution uses area to make estimates of frequencies
(which can be expressed as probabilities). Emphasize that only
some data are well described by a normal distribution.]