PSAT Math Flashcards: Inferences And Claims From Statistics

What this deck covers

This deck focuses on Inferences And Claims From Statistics, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Which claim is supported: A $95\%$ confidence interval for a proportion is $(0.30,0.36)$. Is $0.40$ plausible?

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

Flashcard 2: What condition makes a sample likely to be representative of the population?

Answer: A random sample from the population. Random selection ensures each member has equal chance, avoiding bias.

Flashcard 3: What must be true to generalize a survey result to an entire population?

Answer: The sample must be randomly selected from that population. Only random samples can represent the entire population without bias.

Flashcard 4: What is a confounding variable in a study comparing a treatment and an outcome?

Answer: A variable related to both treatment and outcome that distorts the effect. Affects both variables, making it unclear if treatment truly causes the outcome.

Flashcard 5: What is undercoverage in a sampling frame?

Answer: Some groups in the population are inadequately represented or missing. Sampling frame doesn't include all population members.

Flashcard 6: Identify the correct inference if a poll used a voluntary online response sample.

Answer: Results may be biased; generalization is not reliable. Self-selected samples are biased and not representative.

Flashcard 7: Identify the best conclusion: a $95\%$ CI for $p$ is $[0.42,0.58]$; claim: $p=0.60$.

Answer: The claim $p=0.60$ is not supported by this interval. $0.60$ lies outside the interval, so it's not a plausible value for $p$.

Flashcard 8: What is a placebo effect?

Answer: Response caused by belief in treatment rather than the treatment itself. Psychological response to inactive treatment.

Flashcard 9: What is nonresponse bias in one sentence?

Answer: Nonresponse bias occurs when nonresponders differ systematically from responders. This can make the sample unrepresentative of the full population.

Flashcard 10: Which claim is justified if a poll uses a voluntary online response link?

Answer: The results may be biased and not representative. Self-selection creates bias toward those with strong opinions.

Flashcard 11: What is the main condition needed to generalize a sample result to a population?

Answer: A random sample from the population. Random sampling ensures each member has equal chance of selection, avoiding bias.

Flashcard 12: If a claim is "the mean is $50$" and the interval is $[47,53]$, is the claim plausible?

Answer: Yes, because $50$ is within $[47,53]$. The claimed value falls inside the confidence interval.

Flashcard 13: A $95\%$ confidence interval for a mean is $[12,18]$. What is the point estimate?

Answer: $15$. The point estimate is the interval's midpoint: $(12+18)/2 = 15$.

Flashcard 14: What happens to margin of error when sample size $n$ increases (all else equal)?

Answer: It decreases as $n$ increases. Larger samples provide more information, reducing uncertainty.

Flashcard 15: If two $95\%$ confidence intervals for means do not overlap, what is the likely inference?

Answer: The population means likely differ (evidence of a difference). Non-overlapping CIs suggest statistically significant difference.

Flashcard 16: What is selection bias in a sample?

Answer: The sampling method favors certain outcomes. Some groups are over/underrepresented due to sampling method.

Flashcard 17: Choose the correct claim: A survey of volunteers on a website finds higher exercise rates than average.

Answer: Generalizing to the population is not justified (voluntary response bias). Self-selected volunteers aren't representative of the general population.

Flashcard 18: Identify the correct inference if subjects were randomly assigned but the sample was not random.

Answer: Can infer causation, but cannot generalize to population. Without random sampling, results don't extend beyond the sample.

Flashcard 19: Identify the correct conclusion if a $95\%$ CI for $\mu$ is $[48,56]$ and the claim is $\mu=52$.

Answer: $\mu=52$ is plausible at the $95\%$ level. The CI contains $52$, so that value is plausible.

Flashcard 20: What condition makes a sample mean an unbiased estimator of a population mean?

Answer: A random sample from the population. Random sampling ensures each population member has equal chance of selection.

Flashcard 21: Two $95\%$ confidence intervals for means are $[10,14]$ and $[13,17]$. Do they overlap?

Answer: Yes, they overlap (common values from $13$ to $14$). Intervals overlap when they share at least one common value.

Flashcard 22: Which option best describes a placebo in an experiment?

Answer: A treatment with no active ingredient used for comparison. Controls for psychological effects of receiving treatment.

Flashcard 23: Identify the correct conclusion: A poll of $400$ random voters has $\pm 5\%$ error for a proportion of $52\%$. What interval results?

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

Flashcard 24: What happens to a confidence interval's width when sample size $n$ increases (all else equal)?

Answer: It becomes narrower. Larger samples give more precise estimates with less variability.

Flashcard 25: Identify the correct conclusion if a $95\%$ CI for $p$ is $[0.62,0.70]$ and the claim is $p=0.60$.

Answer: $p=0.60$ is not plausible at the $95\%$ level. The CI excludes $0.60$, so that value is implausible.

Flashcard 26: Choose the correct interpretation if a $95\%$ confidence interval for $p$ is $[0.42,0.58]$.

Answer: The true $p$ is plausibly between $0.42$ and $0.58$. We're $95\%$ confident the interval captures the true proportion.

Flashcard 27: Which statement is correct if a study finds a strong correlation between $x$ and $y$?

Answer: Correlation alone does not imply causation. Correlation measures association, not causal relationship.

Flashcard 28: What is the key condition for a study to support a cause-and-effect claim?

Answer: A randomized experiment with controlled treatment assignment. Random assignment eliminates confounding variables.

Flashcard 29: What is sampling bias in one sentence?

Answer: Sampling bias occurs when the sampling method favors certain outcomes. This creates systematic differences between the sample and population.

Flashcard 30: What sampling problem occurs when some selected individuals cannot be reached or do not respond?

Answer: Nonresponse bias. Missing responses can systematically differ from obtained responses.

Flashcard 31: Which statement is valid when a $95\%$ CI for a proportion includes $0.5$?

Answer: The claim $p=0.5$ is plausible given the data. The CI contains $0.5$, so we can't reject that value.

Flashcard 32: What is the key condition that allows a sample to support inference about a population?

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

Flashcard 33: What type of bias occurs when the sampling method systematically excludes part of the population?

Answer: Undercoverage bias. Some groups have no chance of selection, making the sample unrepresentative.

Flashcard 34: Identify the sampling method: dividing into groups and randomly sampling within each group.

Answer: Stratified random sampling. Ensures representation from each subgroup by sampling within predefined strata.

Flashcard 35: Which option correctly interprets correlation $r=-0.8$ between $x$ and $y$?

Answer: A strong negative linear association (not proof of causation). Correlation measures association strength/direction, never proves causation.

Flashcard 36: What is the purpose of blinding in an experiment?

Answer: To reduce bias from expectations of subjects or researchers. Prevents knowledge of treatment from influencing results.

Flashcard 37: Choose the correct inference: A study finds ice cream sales and drownings both rise. Without an experiment, what is justified?

Answer: Association only; causation is not justified (likely confounding). Both may be caused by a third variable (e.g., summer weather).

Flashcard 38: What does "statistically significant" mean in plain language?

Answer: The result is unlikely due to random chance alone. The observed difference is too large to attribute to chance.

Flashcard 39: Which phrase correctly describes a conclusion from an observational study: "causes" or "is associated with"?

Answer: "Is associated with" (not "causes"). Observational studies can't prove causation due to potential confounding variables.

Flashcard 40: In a scatterplot, what does a correlation near $0$ indicate about linear association?

Answer: Little to no linear relationship. Points show no clear upward or downward trend.

Flashcard 41: What is the definition of a $p$-value?

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

Flashcard 42: What is the most appropriate measure of center for a strongly skewed distribution?

Answer: Median. Median resists influence from extreme values.

Flashcard 43: Find the margin of error for the interval $[18,26]$.

Answer: $4$. Margin of error $= \frac{26-18}{2} = 4$.

Flashcard 44: Identify the most justified statement: two groups differ by $2$ points, but distributions overlap heavily.

Answer: The difference is small relative to variability; the effect may be weak. Large overlap suggests the difference may not be practically significant.

Flashcard 45: Identify the correct conclusion: A poll reports $52\%\pm 4\%$ support. Can you claim a majority supports the measure?

Answer: No; interval includes $50\%$ (from $48\%$ to $56\%$). Interval $[48\%, 56\%]$ contains $50\%$, so majority isn't certain.

Flashcard 46: Identify the correct direction: If sample size $n$ increases, what happens to margin of error?

Answer: It decreases, roughly proportional to $\frac{1}{\sqrt{n}}$. Larger samples give more precise estimates.

Flashcard 47: What is the key condition needed to generalize a sample result to a population?

Answer: A random sample from the target population. Random sampling ensures the sample represents the population without bias.

Flashcard 48: Which option best describes selection bias in a survey sample?

Answer: Some groups are systematically underrepresented. Sampling method excludes or undersamples certain population segments.

Flashcard 49: Which conclusion is valid: "Group A had higher mean than Group B" implies "every A exceeds every B"?

Answer: Invalid; a higher mean does not imply all values are larger. Groups can overlap even when means differ.

Flashcard 50: Identify the correct inference if participants were randomly sampled but not randomly assigned to groups.

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

Flashcard 51: What is a confounding variable in a study comparing two groups?

Answer: A variable related to both the explanatory and response variables. Confounders create false associations by affecting both variables being studied.

Flashcard 52: What is the most common meaning of a $95\%$ confidence interval in context?

Answer: The method captures the true value about $95\%$ of the time. In repeated sampling, $95\%$ of intervals would contain the true parameter.

Flashcard 53: What study type can support a cause-and-effect claim when well designed?

Answer: A randomized controlled experiment. Random assignment of treatments controls for confounding variables.

Flashcard 54: Which sampling method best supports generalizing results to a population?

Answer: Random sampling from the population. Random sampling ensures the sample represents the population without bias.

Flashcard 55: Which conclusion is justified: a random sample supports causation or generalization?

Answer: Generalization to the population, not causation by itself. Random sampling enables inference about populations, not causes.

Flashcard 56: What is the key difference between an observational study and an experiment?

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

Flashcard 57: What is the correct formula for a confidence interval written from estimate and margin of error?

Answer: $\text{estimate}\pm\text{margin of error}$. The interval extends from estimate minus ME to estimate plus ME.

Flashcard 58: What does a $95\%$ confidence interval mean in repeated sampling?

Answer: $95\%$ of such intervals would contain the true parameter. The confidence level refers to the long-run capture rate of the method.

Flashcard 59: What is the margin of error for an estimate based on a confidence interval?

Answer: Half the width of the confidence interval. Margin of error extends equally above and below the point estimate.

Flashcard 60: Identify the flaw: A study finds ice cream sales and drownings rise together, so ice cream causes drownings.

Answer: Correlation does not imply causation; a confounder (season) is likely. Both variables are influenced by a third factor (warm weather).

Flashcard 61: What does a margin of error of $m$ mean for an estimate p?

Answer: A plausible interval is p \pm m. The true value likely falls within $m$ units of the estimate.

Flashcard 62: What type of bias occurs when a question is phrased to influence the respondent's answer?

Answer: Response bias (leading question). Question wording pushes respondents toward a particular answer choice.

Flashcard 63: If a $95\%$ confidence interval for a difference is entirely above $0$, what claim is supported?

Answer: The true difference is positive (evidence that one mean is larger). All values in CI are positive, so first mean exceeds second.

Flashcard 64: What is the most appropriate measure of spread for a distribution with outliers?

Answer: Interquartile range (IQR). IQR uses middle 50%, unaffected by extremes.

Flashcard 65: What is nonresponse bias?

Answer: Nonresponders differ systematically from responders. Missing responses aren't random, creating systematic bias.

Flashcard 66: Identify the best conclusion: a $95\%$ CI for $\mu$ is $[12,18]$; claim: $\mu=15$.

Answer: The claim $\mu=15$ is consistent with this interval. $15$ lies within the interval, so it's a plausible value for $\mu$.

Flashcard 67: What is the key condition for a survey result to generalize to a population?

Answer: A random sample from the target population. Random sampling ensures each member has equal chance of selection.

Flashcard 68: What is the key difference between a sample and a population in statistics?

Answer: Sample is a subset; population is the entire group. Sample is selected from population to make inferences about the whole.

Flashcard 69: Which conclusion is valid: a study uses random assignment but not random sampling?

Answer: You may claim causation, but not generalize to the population. Random assignment allows causal claims; no random sampling limits scope.

Flashcard 70: Which conclusion is valid: a study uses both random sampling and random assignment?

Answer: You may generalize to the population and claim causation. Both conditions are met for causal inference about the population.

Flashcard 71: What happens to margin of error when the sample size $n$ increases?

Answer: It decreases. Larger samples provide more precise estimates with smaller margins.

Flashcard 72: Which claim is valid from an observational study: association or causation?

Answer: Association only, not causation. Without random assignment, confounding variables prevent causal claims.

Flashcard 73: If a confidence interval is $[40,52]$, what is the point estimate (center)?

Answer: $46$. Point estimate is the midpoint: $(40+52)÷2=46$.

Flashcard 74: Identify the correct conclusion: a random sample supports inference about what group?

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

Flashcard 75: If a confidence interval is $[12,18]$, what is the margin of error?

Answer: $3$. Width is $18-12=6$, so margin of error is $6÷2=3$.

Flashcard 76: In a confidence interval, what is the margin of error in terms of endpoints $L$ and $U$?

Answer: $\frac{U-L}{2}$. Half the interval width equals the margin of error.

Flashcard 77: What is a confounding variable?

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

Flashcard 78: What type of study can justify a cause-and-effect claim between two variables?

Answer: A randomized experiment. Only experiments with controlled manipulation can establish causation.

Flashcard 79: What does a $95\%$ confidence level mean in a confidence interval context?

Answer: About $95\%$ of intervals from repeats would capture the true value. If we repeated sampling many times, $95\%$ contain the parameter.

Flashcard 80: What is the meaning of a $95\%$ confidence interval in context?

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

Flashcard 81: What is the correct interpretation of a $95\%$ confidence interval?

Answer: The method captures the true parameter in $95\%$ of repeated samples. The interval construction method, not individual intervals.

Flashcard 82: What type of bias occurs when some selected individuals cannot be contacted or refuse to respond?

Answer: Nonresponse bias. Missing responses may differ systematically from those who do respond.

Flashcard 83: What is the main purpose of random sampling in a study?

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

Flashcard 84: Which conclusion is valid from an observational study: causation or association?

Answer: Association only (not causation). Without random assignment, confounding variables prevent causal claims.

Flashcard 85: What is the key condition needed to generalize a sample result to a population?

Answer: The sample must be randomly selected from the population. Random selection ensures the sample represents the entire population without bias.

Flashcard 86: What is nonresponse bias?

Answer: Bias when nonresponders differ systematically from responders. Creates systematic differences between those who respond and those who don't.

Flashcard 87: Identify the flaw: Surveying only daytime shoppers to estimate all customer satisfaction for a store.

Answer: Selection bias; the sample underrepresents evening and weekend shoppers. Missing key groups leads to unrepresentative results.

Flashcard 88: If intervals from two groups do not overlap, what is a reasonable conclusion about the means?

Answer: The means likely differ (evidence of a difference). Non-overlapping intervals suggest statistically significant difference.

Flashcard 89: What is the key requirement for using a sample proportion to estimate a population proportion?

Answer: The sample must be representative (ideally random). Unbiased sampling ensures the sample reflects the population.

Flashcard 90: Which claim is justified by a random sample but no experiment: causation or association?

Answer: Association (not causation). Observational studies can show relationships but cannot prove one causes the other.

Flashcard 91: What is the margin of error relationship to confidence interval width?

Answer: CI width $= 2\times$ (margin of error). Margin of error extends from center to each endpoint.

Flashcard 92: What condition allows a study to support a cause-and-effect claim between variables?

Answer: Random assignment in a controlled experiment. Only experiments with random assignment can establish causation.

Flashcard 93: Which option best describes a statistically valid comparison in an experiment with two treatments?

Answer: Randomly assign subjects to treatments and keep other conditions the same. Random assignment ensures treatment groups are comparable.

Flashcard 94: Identify the flaw: A poll of website visitors estimates support among all voters.

Answer: Voluntary response bias; the sample is not representative of all voters. Self-selected participants don't represent the broader population.

Flashcard 95: Which claim is valid from a randomized experiment: association or causation?

Answer: Causation (cause-and-effect) may be claimed. Random assignment allows researchers to isolate treatment effects.

Flashcard 96: What is the key condition that allows a study to support a cause-and-effect claim?

Answer: A randomized experiment with controlled treatment assignment. Random assignment of treatments controls for confounding variables.

Flashcard 97: Which option best describes why a larger random sample reduces sampling variability?

Answer: Sample statistics tend to be closer to the population value. Larger samples reduce random variation around the true value.

Flashcard 98: What does a larger sample size generally do to a confidence interval's width?

Answer: It makes the interval narrower. More data reduces sampling variability, creating more precise estimates.

Flashcard 99: What is nonresponse bias in a survey?

Answer: Nonresponders differ systematically from responders. Missing data patterns can skew results.

Flashcard 100: Identify the best conclusion: a $95\%$ interval for $p$ is $[0.38,0.46]$; does it support $p=0.40$?

Answer: Yes; $0.40$ is in $[0.38,0.46]$. Values inside the confidence interval are plausible.