Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 17 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Identify an outlier from a display summary
Interpret differences in shape, center, spread, and outliers in context.
A visual outlier is identified by separation from the main body of data, not merely by being the smallest or largest observation. Locate the dense cluster first, then compare any …
Preview problemCompare distribution shapes
Interpret differences in shape, center, spread, and outliers in context.
Compare distributions along the same shape dimension instead of drifting into center or spread. For each group, ask whether the two sides balance or whether one side stretches into a …
Preview problemCompare centers of two distributions in context
Interpret differences in shape, center, spread, and outliers in context.
A center comparison should connect a numerical difference to what the measured quantity means. Subtract the smaller median from the larger one to get the distance between typical values, then …
Preview problemCompare spreads of two distributions in context
Interpret differences in shape, center, spread, and outliers in context.
Interquartile range describes the width of the middle half, so it compares consistency rather than typical speed. Put the two IQRs side by side, find how much wider one middle …
Preview problemInterpret overlap between two distributions
Interpret differences in shape, center, spread, and outliers in context.
Overlap and center describe different features of two distributions, so measure them separately. Treat each box as an interval, find the common segment and compare it with each box width, …
Preview problemChoose a contextual conclusion supported by display evidence
Interpret differences in shape, center, spread, and outliers in context.
Translate each display feature into only the kind of claim it supports. A median comparison speaks about typical scores, an IQR comparison speaks about consistency in the middle half, and …
Preview problemAssess how an outlier changes interpretation
Interpret differences in shape, center, spread, and outliers in context.
Replacing an extreme value lets you see which statistics depend on distance from the center and which depend mainly on ordered middle positions. Build parallel before-and-after lists, recompute every measure …
Preview problemComplete a two-way frequency table from raw counts
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
Every interior cell in a two-way table represents one row category and one column category happening together. Match both labels before entering a count, then add all interior cells as …
Preview problemFind row totals, column totals, and grand total
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
The margins summarize the same interior counts in two directions. Add across for each row total and down for each column total, then use the agreement between the sum of …
Preview problemCompute a joint relative frequency
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
A joint relative frequency asks what share of the entire table lies in one specific row-and-column intersection. Put that interior cell count in the numerator and the grand total in …
Preview problemCompute a marginal relative frequency
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
A marginal frequency combines across the other variable, so its numerator comes from a row or column total rather than one interior cell. Because the question asks for that margin’s …
Preview problemCompute a row conditional relative frequency
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
Conditioning changes the reference group, and the denominator must follow that change. When the condition names a row, restrict attention to that row, place its total in the denominator, and …
Preview problemCompute a column conditional relative frequency
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
A column condition means the comparison lives entirely inside that column. Use the cell as the part and the column total as the whole, then read the ratio as the …
Preview problemInterpret a joint relative frequency
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
The word joint means both category conditions hold at once, so think intersection rather than union or a conditional subgroup. Name the people satisfying both conditions in the numerator and …
Preview problemInterpret a marginal relative frequency
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
A marginal relative frequency collapses the table across the other variable. Add every cell belonging to the named category, use that margin as the numerator and the grand total as …
Preview problemInterpret a conditional relative frequency
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
Read a conditional statement as “among” the group named after the condition. That group supplies the denominator, while the people in it who also satisfy the outcome supply the numerator; …
Preview problemCompare conditional relative frequencies to describe association
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
To study association, compare the same outcome rate within each group rather than comparing raw counts. Subtract the conditional percentages in percentage points, apply the stated descriptive threshold, and use …
Preview problemIdentify informal independence from conditional frequencies
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
Informal independence is judged by how closely the same conditional rate matches across groups. Compute the absolute percentage-point gap and compare it with the supplied tolerance, then remember that a …
Preview problemChoose the correct denominator for a relative frequency question
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
The denominator comes from the reference group named by words such as “of” or “among,” not from whichever category sounds most prominent. Separate the joint group being counted in the …
Preview problemCreate a total-relative frequency table from counts
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
A total-relative-frequency table puts every cell on one common scale. Find the grand total once, divide each interior count by that same denominator, and verify the finished table by checking …
Preview problemUse two-way table evidence to support or reject a claim
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.
A “more likely” claim across groups is answered by comparing the same within-group percentage for each group. Put the two conditional rates on equal footing, compute their percentage-point gap, and …
Preview problemCreate a scatter plot from paired data
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
A scatter plot preserves each observation as one ordered pair, with the first coordinate on the horizontal axis and the second on the vertical axis. Set labeled windows that contain …
Preview problemIdentify the direction of association in a scatter plot of two quantitative variables
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
Direction asks how the point cloud moves overall as you read from left to right. Track whether typical y-values rise, fall, or show no consistent change as x increases, and …
Preview problemClassify the overall form of a scatter plot as linear, exponential-like, or quadratic-like
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
Form describes the shape of the point cloud, while direction describes whether it rises or falls. Look for a roughly straight band versus a curve whose steepness changes or one …
Preview problemInterpret the strength of association in a scatter plot
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
Direction and strength are two separate readings of a scatter plot. Use the left-to-right movement of the point cloud to describe direction, then judge strength by how tightly the points …
Preview problemIdentify an outlier in a scatter plot
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
A scatter-plot outlier is unusual relative to the dominant relationship, not merely extreme in its x-coordinate. Identify the rule or trend followed by most points, predict where another point with …
Preview problemEstimate a linear model from fit-line points
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
Two well-separated points on a fit line determine its linear model even if they are not original data points. Compute rise over run with a consistent point order, find the …
Preview problemFit an exponential-style model informally
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
For equal input steps, exponential behavior shows up through nearly constant output ratios rather than constant differences. Estimate the common factor, use the output at zero as the initial value, …
Preview problemFit a quadratic-style model informally
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
A change from decreasing to increasing signals a turning point, which a line or a one-direction exponential curve cannot capture. Match that single turn to a quadratic form, then use …
Preview problemUse a fitted model to predict an output from a given input
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
A fitted model turns an input into a predicted output through direct substitution. Replace only the input variable with the requested value, keep every coefficient and constant intact, and evaluate …
Preview problemDecide whether a prediction from a scatter plot model is interpolation or extrapolation
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
Interpolation and extrapolation depend only on where the prediction input sits relative to the observed input domain. Mark the smallest and largest observed inputs, locate the new input on that …
Preview problemChoose the most reasonable model type from a scatter plot or scatter-plot description
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
Model family comes from the shape of the point cloud, while increasing or decreasing only tells direction. A straight band supports a line, changing steepness suggests exponential behavior, and a …
Preview problemInterpret fitted model behavior in context
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
In a linear model, the input coefficient is an additive rate of change, not the entire output or a percent factor. Give it output-units per input-unit, compare model values one …
Preview problemAssess whether a fitted model is appropriate
Use scatter plots to represent two quantitative variables and fit functions to model relationships.
Residuals diagnose whether a model leaves a systematic shape unexplained. Read their signs in input order and look for sustained runs, curvature, or one-sided regions; scatter around zero supports the …
Preview problemCalculate a residual from an observed value and a predicted value
Assess model fit informally using residuals.
A residual measures the signed vertical error of a model, so subtraction order matters. Always compute observed minus predicted; the magnitude tells how far the prediction missed, while the sign …
Preview problemInterpret a residual in context
Assess model fit informally using residuals.
Use the residual equation to connect prediction, observation, and error in one consistent direction. Reconstruct the observation by adding the residual to the prediction, then translate a positive result as …
Preview problem