Inferences & Claims From Statistics - PSAT Math

The population from which the sample was randomly selected. Random sampling allows generalization from sample to population.

Causation (cause-and-effect), not generalization by itself. Random assignment controls confounding, enabling causal conclusions.

$0.08$. Margin of error is half the interval width: $(0.58-0.42)/2 = 0.08$.

$95%$ of intervals from repeated samples contain the true value. If we repeated sampling many times, $95%$ would capture the parameter.

Can generalize to population, but cannot infer causation. Without random assignment, we can't establish causation.

Experiments impose treatments; observational studies do not. Experiments manipulate variables; observational studies only observe.

Confounding variable. It affects both variables, creating a spurious relationship.

A confounding variable could explain the association. Third variables can create non-causal associations.

A cause-and-effect claim about the drug is justified. Random assignment allows causal inference.

It can exaggerate apparent differences between values. Compressed scale magnifies visual differences.

No clear evidence of a difference at the $5%$ level. Zero in interval means difference not statistically significant.

$22$. The center is $\frac{18+26}{2} = 22$.

Median. Median resists the influence of extreme values better than mean.

$\text{IQR}$. IQR measures middle $50%$ spread, unaffected by outliers.

A baseline claim of no effect or no difference, used for comparison. It assumes nothing special is happening, serving as the default position.

The probability of results at least as extreme, assuming the null is true. Smaller $p$-values indicate stronger evidence against the null hypothesis.

Correlation does not imply causation. Two variables can be related without one causing the other.

A variable related to both the explanatory and response variables. It can create false associations between the variables of interest.

$47%$ to $57%$. Add and subtract margin of error: $52% \pm 5%$.

Yes, because $15$ is inside $(12,18)$. Values inside the interval are plausible population values.

No, because $0.40$ is outside $(0.30,0.36)$. Values outside the interval are implausible at this confidence level.

Sampling bias (undercoverage of students not in the cafeteria). Students not in cafeteria aren't represented in the sample.

The tutoring method caused higher scores (causal conclusion). Random assignment allows causal inference about the treatment.

The sample must be random and representative of the population. Random sampling ensures the sample reflects population characteristics.

Parameter describes a population; statistic describes a sample. Parameters are fixed values; statistics vary with samples.

Systematic undercoverage that makes a sample unrepresentative. Occurs when certain groups are excluded or underrepresented.

To support a cause-and-effect conclusion by reducing confounding. Ensures treatment groups are comparable except for the treatment.

To support generalizing results from the sample to the population. Ensures the sample represents the entire population fairly.

A variable related to both the explanatory and response variables. Can create false relationships between variables of interest.