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

Use probability to evaluate outcomes of decisions.

Understand independence and conditional probability and use them to interpret data.

Investigate patterns of association in bivariate data.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Investigate chance processes and develop, use, and evaluate probability models.

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

Draw informal comparative inferences about two populations.

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

Calculate expected values and use them to solve problems.

Use random sampling to draw inferences about a population.

Interpret linear models.

Develop understanding of statistical variability.

Summarize and describe distributions.

Understand and evaluate random processes underlying statistical experiments.
problem_activity: Warm-up
The mathematical purpose of this activity is for students to reason about the area under a normal curve using what they know about composition and decomposition of areas.
If students are using a table of values with
\(z\)
-scores for this lesson, this work will be essential to finding the desired areas in many cases.
Launch
Remind students that the phrase “the area under the curve” means the area between the
\(x\)
-axis and the curve.
Student Facing
The images show a normal curve with mean of 40 and standard deviation of 2.
The area under the curve to the left of 39 is 0.3085.
[IMG]
The area under the curve to the left of 43 is 0.9332.
[IMG]
Since it is a normal curve, we know that the total area under the curve is 1. Use the given areas to find the areas in question. Explain your reasoning for each.
Find the area under the curve to the right of 39.
Find the area under the curve between 39 and 43.
Find the area under the curve to the left of 40.
Find the area under the curve between 39 and 40.
Student Response
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Activity Synthesis
The purpose of this discussion is to make sure students can use the given areas to compose or decompose them to find additional areas. For each question, ask a student to explain how they reasoned about their answer. Highlight places where students used the symmetry of the distribution or cases where they could add or subtract known areas to find new areas.


correct_option_index: 6