Practice Hidden Variables in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Quantities or factors that influence a mathematical or real-world situation but are not explicitly included in the current model or expression.
What's lurking behind the scenes that we forgot to account for?
Showing a random 20 of 50 problems.
Example 1
mediumA linear fit of y on x has slope 2. After adding an omitted variable z (correlated with both), the slope of x drops to 0.5. What does this reveal about the original model?
Example 2
challengeIn a regression y = b0 + b1 x, the true model is y = b0 + b1 x + b2 z + e, with z = c x + u. Show that omitting z makes the estimated x-coefficient converge to b1 + b2 c, and state the condition under which omission causes no bias.
Example 3
easyA store finds taller shelves sell more, so stocks taller shelves. Sales drop. What hidden variable might have driven the original pattern?
Example 4
challengeAcross 5 years, a treatment's yearly success rate exceeds control's every year, yet pooled over all years control wins. Construct the hidden-variable condition (counts) that makes this possible and name the phenomenon.
Example 5
challengeTwo regions show the same average household income , but region A has a Gini of and region B has . What hidden variable does a single mean conceal?
Example 6
mediumA linear model holds in summer. In winter, the same data show . Identify the hidden variable and write the augmented model.
Example 7
mediumRegion A and Region B both show a hospital treatment with 80% survival overall, yet within both mild and severe cases Hospital X beats Hospital Y. Which hidden variable produces this reversal (Simpson's paradox)?
Example 8
easyThe area formula has hidden units. If m and m, what is , and what hidden variable (units) must be tracked?
Example 9
easyFill in the blank: A variable that influences a relationship but is not included in the model is called a ____ variable.
Example 10
mediumA simple regression fits perfectly on training data with . Adding a measured variable yields with . Explain what was hidden.
Example 11
hardA regression of wages on years of education produces slope . After adding IQ, the slope drops to . Compute the share of the original slope attributable to IQ-correlated effects.
Example 12
easyA formula gives a car's stopping distance as (metres, km/h). Identify the hidden variables that this formula ignores.
Example 13
mediumTwo students score identically on a test, but one cheated. A model using only score predicts equal ability. What hidden variable invalidates the prediction?
Example 14
easyA formula models a falling object's time using only height, ignoring air. For a feather it fails badly. What hidden variable matters?
Example 15
hardA disease has prevalence ; a test has sensitivity and specificity . Compute the posterior probability of disease given a positive test.
Example 16
mediumA 2x2 table compares treatment A vs B by gender. Within each gender A wins; pooled, B wins. Which hidden variable produces the reversal?
Example 17
mediumA regression coefficient changes when a new predictor is added. What does this typically indicate?
Example 18
mediumA test for a rare disease (1% prevalence) is 90% accurate. A patient tests positive. The 'hidden' factor inflating false positives is base rate. What is the approximate probability the patient actually has the disease?
Example 19
mediumA company finds employees with bigger offices earn more, and concludes office size raises pay. Name the most likely hidden variable and the true causal direction.
Example 20
hardA test reports drug A is better in adults and in children separately, but worse overall. Construct counts (adult 90/100 vs 60/100, child 10/100 vs 30/100 for A vs B) and confirm.