Inferences & Claims From Statistics - PSAT Math

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

No, because $0.40$ is outside $(0.30,0.36)$ . Values outside the interval are implausible at this confidence level.
What condition makes a sample likely to be representative of the population?

A random sample from the population. Random selection ensures each member has equal chance, avoiding bias.
What must be true to generalize a survey result to an entire population?

The sample must be randomly selected from that population. Only random samples can represent the entire population without bias.
What is a confounding variable in a study comparing a treatment and an outcome?

A variable related to both treatment and outcome that distorts the effect. Affects both variables, making it unclear if treatment truly causes the outcome.
What is undercoverage in a sampling frame?

Some groups in the population are inadequately represented or missing. Sampling frame doesn't include all population members.
Identify the correct inference if a poll used a voluntary online response sample.

Results may be biased; generalization is not reliable. Self-selected samples are biased and not representative.
Identify the best conclusion: a $95\%$ CI for $p$ is $[0.42,0.58]$ ; claim: $p=0.60$ .

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 $p$ .
What is a placebo effect?

Response caused by belief in treatment rather than the treatment itself. Psychological response to inactive treatment.
What is nonresponse bias in one sentence?

Nonresponse bias occurs when nonresponders differ systematically from responders. This can make the sample unrepresentative of the full population.
Which claim is justified if a poll uses a voluntary online response link?

The results may be biased and not representative. Self-selection creates bias toward those with strong opinions.
What is the main condition needed to generalize a sample result to a population?

A random sample from the population. Random sampling ensures each member has equal chance of selection, avoiding bias.
If a claim is "the mean is $50$ " and the interval is $[47,53]$ , is the claim plausible?

Yes, because $50$ is within $[47,53]$ . The claimed value falls inside the confidence interval.
A $95\%$ confidence interval for a mean is $[12,18]$ . What is the point estimate?

$15$ . The point estimate is the interval's midpoint: $(12+18)/2 = 15$ .
What happens to margin of error when sample size $n$ increases (all else equal)?

It decreases as $n$ increases. Larger samples provide more information, reducing uncertainty.
If two $95\%$ confidence intervals for means do not overlap, what is the likely inference?

The population means likely differ (evidence of a difference). Non-overlapping CIs suggest statistically significant difference.
What is selection bias in a sample?

The sampling method favors certain outcomes. Some groups are over/underrepresented due to sampling method.
Choose the correct claim: A survey of volunteers on a website finds higher exercise rates than average.

Generalizing to the population is not justified (voluntary response bias). Self-selected volunteers aren't representative of the general population.
Identify the correct inference if subjects were randomly assigned but the sample was not random.

Can infer causation, but cannot generalize to population. Without random sampling, results don't extend beyond the sample.
Identify the correct conclusion if a $95\%$ CI for $\mu$ is $[48,56]$ and the claim is $\mu=52$ .

$\mu=52$ is plausible at the $95\%$ level. The CI contains $52$ , so that value is plausible.
What condition makes a sample mean an unbiased estimator of a population mean?

A random sample from the population. Random sampling ensures each population member has equal chance of selection.
Two $95\%$ confidence intervals for means are $[10,14]$ and $[13,17]$ . Do they overlap?

Yes, they overlap (common values from $13$ to $14$ ). Intervals overlap when they share at least one common value.
Which option best describes a placebo in an experiment?

A treatment with no active ingredient used for comparison. Controls for psychological effects of receiving treatment.
Identify the correct conclusion: A poll of $400$ random voters has $\pm 5\%$ error for a proportion of $52\%$ . What interval results?

$47\%$ to $57\%$ . Add and subtract margin of error: $52\% \pm 5\%$ .
What happens to a confidence interval's width when sample size $n$ increases (all else equal)?

It becomes narrower. Larger samples give more precise estimates with less variability.
Identify the correct conclusion if a $95\%$ CI for $p$ is $[0.62,0.70]$ and the claim is $p=0.60$ .

$p=0.60$ is not plausible at the $95\%$ level. The CI excludes $0.60$ , so that value is implausible.
Choose the correct interpretation if a $95\%$ confidence interval for $p$ is $[0.42,0.58]$ .

The true $p$ is plausibly between $0.42$ and $0.58$ . We're $95\%$ confident the interval captures the true proportion.
Which statement is correct if a study finds a strong correlation between $x$ and $y$ ?

Correlation alone does not imply causation. Correlation measures association, not causal relationship.
What is the key condition for a study to support a cause-and-effect claim?

A randomized experiment with controlled treatment assignment. Random assignment eliminates confounding variables.
What is sampling bias in one sentence?

Sampling bias occurs when the sampling method favors certain outcomes. This creates systematic differences between the sample and population.
What sampling problem occurs when some selected individuals cannot be reached or do not respond?

Nonresponse bias. Missing responses can systematically differ from obtained responses.
Which statement is valid when a $95\%$ CI for a proportion includes $0.5$ ?

The claim $p=0.5$ is plausible given the data. The CI contains $0.5$ , so we can't reject that value.
What is the key condition that allows a sample to support inference about a population?

The sample must be random and representative of the population. Random sampling ensures the sample reflects population characteristics.
What type of bias occurs when the sampling method systematically excludes part of the population?

Undercoverage bias. Some groups have no chance of selection, making the sample unrepresentative.
Identify the sampling method: dividing into groups and randomly sampling within each group.

Stratified random sampling. Ensures representation from each subgroup by sampling within predefined strata.
Which option correctly interprets correlation $r=-0.8$ between $x$ and $y$ ?

A strong negative linear association (not proof of causation). Correlation measures association strength/direction, never proves causation.
What is the purpose of blinding in an experiment?

To reduce bias from expectations of subjects or researchers. Prevents knowledge of treatment from influencing results.
Choose the correct inference: A study finds ice cream sales and drownings both rise. Without an experiment, what is justified?

Association only; causation is not justified (likely confounding). Both may be caused by a third variable (e.g., summer weather).
What does "statistically significant" mean in plain language?

The result is unlikely due to random chance alone. The observed difference is too large to attribute to chance.
Which phrase correctly describes a conclusion from an observational study: "causes" or "is associated with"?

"Is associated with" (not "causes"). Observational studies can't prove causation due to potential confounding variables.
In a scatterplot, what does a correlation near $0$ indicate about linear association?

Little to no linear relationship. Points show no clear upward or downward trend.
What is the definition of a $p$ -value?

The probability of results at least as extreme, assuming the null is true. Smaller $p$ -values indicate stronger evidence against the null hypothesis.
What is the most appropriate measure of center for a strongly skewed distribution?

Median. Median resists influence from extreme values.
Find the margin of error for the interval $[18,26]$ .

$4$ . Margin of error $= \frac{26-18}{2} = 4$ .
Identify the most justified statement: two groups differ by $2$ points, but distributions overlap heavily.

The difference is small relative to variability; the effect may be weak. Large overlap suggests the difference may not be practically significant.
Identify the correct conclusion: A poll reports $52\%\pm 4\%$ support. Can you claim a majority supports the measure?

No; interval includes $50\%$ (from $48\%$ to $56\%$ ). Interval $[48\%, 56\%]$ contains $50\%$ , so majority isn't certain.
Identify the correct direction: If sample size $n$ increases, what happens to margin of error?

It decreases, roughly proportional to $\frac{1}{\sqrt{n}}$ . Larger samples give more precise estimates.
What is the key condition needed to generalize a sample result to a population?

A random sample from the target population. Random sampling ensures the sample represents the population without bias.
Which option best describes selection bias in a survey sample?

Some groups are systematically underrepresented. Sampling method excludes or undersamples certain population segments.
Which conclusion is valid: "Group A had higher mean than Group B" implies "every A exceeds every B"?

Invalid; a higher mean does not imply all values are larger. Groups can overlap even when means differ.
Identify the correct inference if participants were randomly sampled but not randomly assigned to groups.

Can generalize to population, but cannot infer causation. Without random assignment, we can't establish causation.
What is a confounding variable in a study comparing two groups?

A variable related to both the explanatory and response variables. Confounders create false associations by affecting both variables being studied.
What is the most common meaning of a $95\%$ confidence interval in context?

The method captures the true value about $95\%$ of the time. In repeated sampling, $95\%$ of intervals would contain the true parameter.
What study type can support a cause-and-effect claim when well designed?

A randomized controlled experiment. Random assignment of treatments controls for confounding variables.
Which sampling method best supports generalizing results to a population?

Random sampling from the population. Random sampling ensures the sample represents the population without bias.
Which conclusion is justified: a random sample supports causation or generalization?

Generalization to the population, not causation by itself. Random sampling enables inference about populations, not causes.
What is the key difference between an observational study and an experiment?

Experiments impose treatments; observational studies do not. Experiments manipulate variables; observational studies only observe.
What is the correct formula for a confidence interval written from estimate and margin of error?

$\text{estimate}\pm\text{margin of error}$ . The interval extends from estimate minus ME to estimate plus ME.
What does a $95\%$ confidence interval mean in repeated sampling?

$95\%$ of such intervals would contain the true parameter. The confidence level refers to the long-run capture rate of the method.
What is the margin of error for an estimate based on a confidence interval?

Half the width of the confidence interval. Margin of error extends equally above and below the point estimate.
Identify the flaw: A study finds ice cream sales and drownings rise together, so ice cream causes drownings.

Correlation does not imply causation; a confounder (season) is likely. Both variables are influenced by a third factor (warm weather).
What does a margin of error of $m$ mean for an estimate $p$ ?

A plausible interval is $p \pm m$ . The true value likely falls within $m$ units of the estimate.
What type of bias occurs when a question is phrased to influence the respondent's answer?

Response bias (leading question). Question wording pushes respondents toward a particular answer choice.
If a $95\%$ confidence interval for a difference is entirely above $0$ , what claim is supported?

The true difference is positive (evidence that one mean is larger). All values in CI are positive, so first mean exceeds second.
What is the most appropriate measure of spread for a distribution with outliers?

Interquartile range (IQR). IQR uses middle 50%, unaffected by extremes.
What is nonresponse bias?

Nonresponders differ systematically from responders. Missing responses aren't random, creating systematic bias.
Identify the best conclusion: a $95\%$ CI for $\mu$ is $[12,18]$ ; claim: $\mu=15$ .

The claim $\mu=15$ is consistent with this interval. $15$ lies within the interval, so it's a plausible value for $\mu$ .
What is the key condition for a survey result to generalize to a population?

A random sample from the target population. Random sampling ensures each member has equal chance of selection.
What is the key difference between a sample and a population in statistics?

Sample is a subset; population is the entire group. Sample is selected from population to make inferences about the whole.
Which conclusion is valid: a study uses random assignment but not random sampling?

You may claim causation, but not generalize to the population. Random assignment allows causal claims; no random sampling limits scope.
Which conclusion is valid: a study uses both random sampling and random assignment?

You may generalize to the population and claim causation. Both conditions are met for causal inference about the population.
What happens to margin of error when the sample size $n$ increases?

It decreases. Larger samples provide more precise estimates with smaller margins.
Which claim is valid from an observational study: association or causation?

Association only, not causation. Without random assignment, confounding variables prevent causal claims.
If a confidence interval is $[40,52]$ , what is the point estimate (center)?

$46$ . Point estimate is the midpoint: $(40+52)÷2=46$ .
Identify the correct conclusion: a random sample supports inference about what group?

The population from which the sample was randomly selected. Random sampling allows generalization from sample to population.
If a confidence interval is $[12,18]$ , what is the margin of error?

$3$ . Width is $18-12=6$ , so margin of error is $6÷2=3$ .
In a confidence interval, what is the margin of error in terms of endpoints $L$ and $U$ ?

$\frac{U-L}{2}$ . Half the interval width equals the margin of error.
What is a confounding variable?

A variable related to both the explanatory and response variables. Can create false relationships between variables of interest.
What type of study can justify a cause-and-effect claim between two variables?

A randomized experiment. Only experiments with controlled manipulation can establish causation.
What does a $95\%$ confidence level mean in a confidence interval context?

About $95\%$ of intervals from repeats would capture the true value. If we repeated sampling many times, $95\%$ contain the parameter.
What is the meaning of a $95\%$ confidence interval in context?

$95\%$ of intervals from repeated samples contain the true value. If we repeated sampling many times, $95\%$ would capture the parameter.
What is the correct interpretation of a $95\%$ confidence interval?

The method captures the true parameter in $95\%$ of repeated samples. The interval construction method, not individual intervals.
What type of bias occurs when some selected individuals cannot be contacted or refuse to respond?

Nonresponse bias. Missing responses may differ systematically from those who do respond.
What is the main purpose of random sampling in a study?

To support generalizing results from the sample to the population. Ensures the sample represents the entire population fairly.
Which conclusion is valid from an observational study: causation or association?

Association only (not causation). Without random assignment, confounding variables prevent causal claims.
What is the key condition needed to generalize a sample result to a population?

The sample must be randomly selected from the population. Random selection ensures the sample represents the entire population without bias.
What is nonresponse bias?

Bias when nonresponders differ systematically from responders. Creates systematic differences between those who respond and those who don't.
Identify the flaw: Surveying only daytime shoppers to estimate all customer satisfaction for a store.

Selection bias; the sample underrepresents evening and weekend shoppers. Missing key groups leads to unrepresentative results.
If intervals from two groups do not overlap, what is a reasonable conclusion about the means?

The means likely differ (evidence of a difference). Non-overlapping intervals suggest statistically significant difference.
What is the key requirement for using a sample proportion to estimate a population proportion?

The sample must be representative (ideally random). Unbiased sampling ensures the sample reflects the population.
Which claim is justified by a random sample but no experiment: causation or association?

Association (not causation). Observational studies can show relationships but cannot prove one causes the other.
What is the margin of error relationship to confidence interval width?

CI width $= 2\times$ (margin of error). Margin of error extends from center to each endpoint.
What condition allows a study to support a cause-and-effect claim between variables?

Random assignment in a controlled experiment. Only experiments with random assignment can establish causation.
Which option best describes a statistically valid comparison in an experiment with two treatments?

Randomly assign subjects to treatments and keep other conditions the same. Random assignment ensures treatment groups are comparable.
Identify the flaw: A poll of website visitors estimates support among all voters.

Voluntary response bias; the sample is not representative of all voters. Self-selected participants don't represent the broader population.
Which claim is valid from a randomized experiment: association or causation?

Causation (cause-and-effect) may be claimed. Random assignment allows researchers to isolate treatment effects.
What is the key condition that allows a study to support a cause-and-effect claim?

A randomized experiment with controlled treatment assignment. Random assignment of treatments controls for confounding variables.
Which option best describes why a larger random sample reduces sampling variability?

Sample statistics tend to be closer to the population value. Larger samples reduce random variation around the true value.
What does a larger sample size generally do to a confidence interval's width?

It makes the interval narrower. More data reduces sampling variability, creating more precise estimates.
What is nonresponse bias in a survey?

Nonresponders differ systematically from responders. Missing data patterns can skew results.
Identify the best conclusion: a $95\%$ interval for $p$ is $[0.38,0.46]$ ; does it support $p=0.40$ ?

Yes; $0.40$ is in $[0.38,0.46]$ . Values inside the confidence interval are plausible.
Which option is a valid inference from a random sample: population conclusion or only sample conclusion?

Population conclusion (with uncertainty), because the sample is random. Random sampling allows generalization from sample to population.
What is the key condition needed to claim a treatment causes an effect?

A randomized experiment with controlled assignment. Random assignment eliminates confounding variables to establish cause-effect.
Which claim is NOT justified by correlation alone, even if the correlation is strong?

A change in $x$ causes a change in $y$ . Correlation doesn't imply causation, even when strong.
What is selection bias in sampling?

A sampling method that systematically favors certain outcomes. Occurs when sampling procedure doesn't give all members equal chance.
What is the margin of error if a $95\%$ confidence interval is $0.42$ to $0.58$ ?

$0.08$ . Margin of error is half the interval width: $(0.58-0.42)/2 = 0.08$ .
What is the formula for the sample mean of $n$ values $x_1,\dots,x_n$ ?

$\bar{x}=\frac{x_1+\cdots+x_n}{n}$ . Sum all values and divide by the count.
Choose the correct conclusion: $62\%\pm 3\%$ support a policy; does this support a majority claim?

Yes; the interval is above $50\%$ (from $59\%$ to $65\%$ ). Entire confidence interval exceeds $50\%$ , confirming majority.
What is the correct rule for deciding if two means are significantly different using CIs?

Non-overlapping CIs suggest a significant difference. If intervals don't overlap, the difference is statistically significant.
If a graph uses a truncated $y$ -axis (not starting at $0$ ), what is the likely effect?

It can exaggerate apparent differences between values. Compressed scale magnifies visual differences.
Which statement is correct: correlation implies causation, or correlation does not imply causation?

Correlation does not imply causation. Association can exist without one variable causing the other.
What does it mean if a study is observational rather than experimental?

Researchers observe outcomes without assigning treatments. No manipulation occurs; researchers simply measure existing conditions.
Which claim is best supported by a $95\%$ confidence interval entirely above $0$ for $p_1-p_2$ ?

There is evidence that $p_1>p_2$ . All positive values mean $p_1-p_2>0$ , so $p_1>p_2$ .
If a $p$ -value is $0.03$ and the significance level is $0.05$ , what is the decision?

Reject the null hypothesis. When $p$ -value $< \alpha$ , we have significant evidence against the null.
What is the meaning of the margin of error in an estimate?

The maximum likely distance between the estimate and true value. Represents uncertainty in the estimate.
Identify the point estimate for a population proportion $p$ from a sample of size $n$ .

$\hat{p}=\frac{x}{n}$ . Sample proportion equals successes divided by sample size.
Choose the correct interpretation: A $90\%$ confidence interval for $p$ is $[0.20,0.30]$ .

The true proportion $p$ is plausibly between $0.20$ and $0.30$ . We're 90% confident the parameter lies in this interval.
What condition allows a study to generalize results to a population from a sample?

Random sampling from the target population. Random sampling ensures the sample represents the population.
Identify the correct interpretation of correlation $r=-0.8$ between $x$ and $y$ .

A strong negative linear association. Negative $r$ near $-1$ indicates strong inverse linear relationship.
Identify the correct claim if a survey uses a voluntary online poll to estimate a population mean.

Generalization to the population is not justified due to likely bias. Self-selection creates bias, preventing valid generalization.
What does " $95\%$ confidence" mean for a method that builds confidence intervals?

About $95\%$ of intervals from repeated samples contain the true value. The method captures the true parameter in $95\%$ of samples.
Which is more precise: margin of error $0.03$ or $0.08$ ?

Margin of error $0.03$ is more precise. Smaller margin of error means narrower interval, higher precision.
What is a double-blind experiment designed to prevent?

Bias from subjects and researchers knowing treatment assignments. Knowledge of treatment can influence behavior and outcomes.
Identify the likely issue: a poll asks, "Do you agree the unfair policy should be ended?"

Wording bias (leading question). Using "unfair" prejudices responses toward ending the policy.
What study type is required to generalize results to a population from a sample?

A random sample from the target population. Random sampling ensures the sample represents the population without bias.
Choose the correct inference: An experiment randomly assigns a new tutoring method and finds higher scores. What can be concluded?

The tutoring method caused higher scores (causal conclusion). Random assignment allows causal inference about the treatment.
Identify the correct conclusion: subjects are randomly assigned to drug or placebo groups.

A cause-and-effect claim about the drug is justified. Random assignment allows causal inference.
What does it mean if a result is statistically significant at the $5\%$ level?

The result is unlikely if the null model is true (about $<5\%$ chance). The result would rarely occur by chance alone under the null hypothesis.
Which phrase best describes a result from an observational study: "causes" or "is associated with"?

"Is associated with". Observational studies can't prove causation, only correlation.
What study type can show association but not causation?

An observational study. Without random assignment, confounding variables may explain observed relationships.
What is the purpose of blinding in an experiment?

To reduce bias from expectations of subjects or researchers. Prevents knowledge of treatment from influencing behavior or measurements.
What is the correct interpretation of a $95\%$ confidence interval for a parameter?

The method captures the true value about $95\%$ of the time. If we repeated sampling many times, 95% of intervals would contain the parameter.
Identify the sampling method: randomly selecting entire groups, then surveying everyone in them.

Cluster sampling. Selects whole groups randomly, then includes all members within chosen clusters.
What is a confounding variable in an observational study?

A variable related to both the explanatory and response variables. It can create false associations between the variables of interest.
What happens to the margin of error when the sample size $n$ increases?

It decreases (typically proportional to $\frac{1}{\sqrt{n}}$ ). Larger samples give more precise estimates.
What is the correct rule for deciding if two proportions are significantly different using CIs?

Non-overlapping CIs suggest a significant difference. Same rule applies for proportions as for means.
Identify the correct claim if a scatterplot shows a strong positive linear trend.

There is a positive association between the variables. Positive trend means variables increase together, but not necessarily causally.
What happens to typical margin of error when confidence level increases (all else equal)?

It increases. Higher confidence requires wider intervals to capture the true value.
Which conclusion is justified: random assignment in an experiment supports causation or generalization?

Causation (cause-and-effect), not generalization by itself. Random assignment controls confounding, enabling causal conclusions.
Find the interval: estimate $=50$ and margin of error $=3$ .

$[47,53]$ . $50-3=47$ and $50+3=53$ give the interval endpoints.
What is the correct interpretation of a $95\%$ confidence interval in repeated sampling?

$95\%$ of such intervals would capture the true parameter. In repeated sampling, 95% of intervals contain the true value.
Which conclusion is valid from a randomized experiment: causation or association?

Causation (and therefore association). Random assignment allows causal inference; causation implies association.
What is the key condition for a sample statistic to support a claim about a population?

The sample must be random and representative of the population. Without randomness, the sample may not reflect the population's characteristics.
What does it mean if a result is statistically significant at level $0.05$ ?

Observed result is unlikely if the null claim is true. P-value < 0.05 means less than 5% chance of seeing this result by random chance alone.
Which conclusion is justified: association or causation, when a study is observational only?

Association only, not causation. Without random assignment, confounding prevents causal claims.
If a claim is "the proportion is $0.60$ " and the interval is $[0.52,0.58]$ , is it supported?

No, because $0.60$ is not within $[0.52,0.58]$ . The claimed value falls outside the confidence interval.
Which option best describes a placebo effect in an experiment?

Change due to belief in treatment, not the treatment itself. Psychological response independent of actual treatment.
What is the main condition needed to claim a treatment caused an effect?

Random assignment to treatment and control groups. Random assignment eliminates confounding variables between groups.
What is selection bias in a sampling method?

Some population members are less likely to be sampled than others. Unequal sampling probability creates systematic error.
Which claim is best supported if a $95\%$ confidence interval for $p_1-p_2$ contains $0$ ?

No clear evidence of a difference between the proportions. Zero in the interval means the difference could be zero.
What is a confounding variable in a study comparing a treatment and an outcome?

A variable related to both treatment and outcome. Confounders create spurious relationships by affecting both variables.
Which option is the best measure of center for skewed data with outliers: mean or median?

Median. Median resists the pull of extreme values better than mean.
Find the margin of error if a $90\%$ interval is $12.4$ to $15.6$ .

$1.6$ . MOE = $\frac{15.6-12.4}{2} = \frac{3.2}{2} = 1.6$ .
What is the key condition that allows a sample to support a claim about a population?

The sample must be random and representative of the population. Without randomness and representativeness, the sample may not reflect the population.
What is the correct inference if a sample statistic differs from the population parameter?

Sampling variability can cause the difference. Random samples naturally vary from population values due to chance.
Which statement is valid: correlation implies causation, or correlation does not imply causation?

Correlation does not imply causation. Two variables can be related without one causing the other.
What is the sample proportion if $37$ out of $100$ surveyed students said yes?

$0.37$ . Sample proportion equals successes divided by sample size: $37/100$ .
If a $95\%$ confidence interval for a mean difference excludes $0$ , what is the inference?

There is evidence the true mean difference is not $0$ . Zero outside interval suggests statistically significant difference.
What does a $95\%$ confidence interval for a mean represent in context?

A plausible range for the population mean based on the sample. We're $95\%$ confident the true mean lies in this interval.
What is the main purpose of random assignment in an experiment?

To support a cause-and-effect conclusion by reducing confounding. Ensures treatment groups are comparable except for the treatment.
Which sample is more likely representative: voluntary response or simple random sample?

Simple random sample. Random sampling avoids bias; voluntary response attracts extremes.
Which conclusion is valid: a study uses random sampling but no random assignment?

You may generalize to the population, but not claim causation. Random sampling allows generalization; no manipulation means no causation.
Identify the sampling method: selecting every $10$ th name from a complete list.

Systematic sampling. Selecting at regular intervals from an ordered list defines systematic sampling.
What is the main difference between correlation and causation?

Correlation is association; causation requires a cause-and-effect link. Correlation shows relationship; causation proves one variable causes the other.
Identify the best critique: A school surveys only students in the cafeteria to estimate schoolwide lunch opinions. What issue occurs?

Sampling bias (undercoverage of students not in the cafeteria). Students not in cafeteria aren't represented in the sample.
If a $p$ -value is $0.20$ and the significance level is $0.05$ , what is the decision?

Fail to reject the null hypothesis. When $p$ -value $> \alpha$ , we lack sufficient evidence against the null.
What is the most common sampling issue when only volunteers respond to a survey?

Voluntary response bias. Self-selected participants often have stronger opinions than the general population.
Which option is a valid generalization: A random sample is taken from a town, but no random assignment is used.

You may generalize to the town, but you cannot infer causation. Random sampling allows generalization; no random assignment means no causal claims.
Identify the sampling method: selecting every $10$ th name on an alphabetized list of all students.

Systematic sampling. Selects at regular intervals from an ordered list.
What is the key condition for making a causal claim from a study's results?

A randomized experiment with controlled conditions. Only experiments can establish causation; observational studies show correlation.
What is the definition of a margin of error for an estimate?

A maximum likely distance from the true value. Quantifies uncertainty by giving the furthest plausible deviation.
Which statement is justified: A randomized experiment shows the treatment group mean exceeds control by $10$ units.

The treatment likely caused an increase (for this study). Random assignment allows causal inference from observed difference.
Which conclusion is justified: association or causation, when an experiment uses random assignment?

Causation can be concluded (for the experimental setting). Random assignment controls confounding variables.
What is the margin of error for an interval estimate written as estimate $\pm$ MOE?

The distance from the estimate to either endpoint. MOE is half the interval width: $\frac{\text{upper}-\text{lower}}{2}$ .
Identify the error: A survey of $30$ students in one class is used to infer all students in the state.

Sample is not representative of the target population. One class cannot represent all students statewide.
Which conclusion is valid: " $A$ causes $B$ " or " $A$ is associated with $B$ ," given an observational study?

" $A$ is associated with $B$ ". Without randomization, only association can be claimed.
Which statement is correct: A larger confidence level gives a wider or narrower interval?

Wider interval. Higher confidence requires capturing more possible values.
What is nonresponse bias?

Bias when selected individuals do not respond and differ systematically. Non-responders may differ from responders in important ways.
What is the difference between a parameter and a statistic?

Parameter describes a population; statistic describes a sample. Parameters are fixed values; statistics vary with samples.
Which claim is supported: A $90\%$ confidence interval for a mean is $(12,18)$ . Is $15$ a plausible population mean?

Yes, because $15$ is inside $(12,18)$ . Values inside the interval are plausible population values.
Which sampling method best supports generalizing to a population: convenience sample or random sample?

Random sample. Only random samples represent the population without bias.
Choose the correct interpretation: " $p=0.03$ means the null hypothesis is true with probability $0.03$ ." True or false?

False. $p$ -value is probability of data given null is true, not vice versa.
What is a leading question in a survey?

A question worded to push respondents toward a particular answer. Influences responses through suggestive or biased wording.
What is blinding in an experiment intended to reduce?

Bias from participants' or researchers' expectations. Prevents knowledge from influencing outcomes or measurements.
Which term describes a variable that influences both the explanatory and response variables?

Confounding variable. It affects both variables, creating a spurious relationship.
Which sampling method is described: divide into groups and randomly sample within each group?

Stratified random sample. Ensures representation from each subgroup (stratum).
What does it mean if a result is statistically significant at the $5\%$ level?

Such a result is unlikely by chance alone if the null claim is true. Less than $5\%$ probability of occurring by random chance.
Which statement is valid when a $95\%$ CI for a mean difference includes $0$ ?

No statistically significant difference is supported. Zero in the CI means no difference is plausible.
Which claim is justified by a randomized experiment on volunteers: causation or generalization?

Causation (not generalization to a population). Without random sampling from a population, results apply only to study participants.
Identify the sampling method: Surveying only people who choose to click an online poll link.

Voluntary response sample (likely biased). Self-selected respondents don't represent the full population.
What is the definition of a confounding variable in a statistical study?

A variable related to both explanatory and response variables. Confounders affect both variables, creating spurious relationships.
Which comparison is correct: Two estimates are $40\%\pm 2\%$ and $44\%\pm 2\%$ . Do intervals overlap?

No; $[38\%,42\%]$ and $[42\%,46\%]$ only meet at $42\%$ . First interval: $[38\%, 42\%]$ ; second: $[42\%, 46\%]$ ; they touch at exactly $42\%$ .
Identify the correct conclusion: " $r=0.8$ implies one variable causes the other." True or false?

False; correlation does not imply causation. Strong correlation indicates association, not causation.
Identify the more appropriate spread for skewed data with outliers: standard deviation or $\text{IQR}$ ?

$\text{IQR}$ . IQR measures middle $50\%$ spread, unaffected by outliers.
What is the placebo effect in an experiment?

Change due to belief in treatment, not the treatment itself. Psychological improvement from expecting treatment to work, not actual treatment.
What does a higher confidence level generally do to a confidence interval's width?

It makes the interval wider. Higher confidence requires capturing more possible values of the parameter.
Identify the correct inference: A random sample of $200$ voters gives $54\%\pm 3\%$ support. What population support is reasonable?

About $51\%$ to $57\%$ . The interval $54\% \pm 3\%$ gives range $51\%$ to $57\%$ .
What does a margin of error represent in a confidence interval?

The maximum likely distance from the estimate to the true value. It's half the width of the confidence interval.
What is the definition of sampling bias in a statistical study?

Some population members are underrepresented in the sample. Bias occurs when the sample doesn't fairly represent all groups.
Which sampling method is described: select every $10$ th name on an ordered list?

Systematic sample. Fixed interval selection from an ordered list.
If a $95\%$ confidence interval for a proportion difference includes $0$ , what is the inference?

No clear evidence of a difference at the $5\%$ level. Zero in interval means difference not statistically significant.
If the margin of error is $4$ and the point estimate is $30$ , what is the interval?

$[26,34]$ . Add and subtract margin of error from point estimate: $30±4$ .
Which option best describes correlation in a scatterplot with points near a rising line?

Strong positive association. Points clustered around an upward-sloping line show positive correlation.
Which claim is justified: random assignment to treatments or random sampling from a population?

Random assignment supports causation; random sampling supports generalization. Assignment controls variables; sampling represents populations.
What does it mean if a statistic is "statistically significant" at a stated level?

The result is unlikely to be due to random chance alone. The observed difference is too large to attribute to sampling variability.
Identify the correct conclusion: A random sample shows $62\%$ support with $\pm 3\%$ margin of error.

Support is about $59\%$ to $65\%$ in the population. Margin of error gives the range: $62\% \pm 3\% = 59\%$ to $65\%$ .
What does it mean if a sample is biased?

Some outcomes are systematically favored in selection. Bias occurs when the sampling method doesn't give all members equal chance.
What condition makes a sample statistic an unbiased estimator of a population parameter?

Random sampling from the target population. Ensures each population member has equal chance of selection, avoiding systematic bias.
What is selection bias in a sample or survey?

Some population members are systematically more likely to be sampled. Unequal sampling probability creates a sample that doesn't represent the population.
What is the null hypothesis (null model) in a statistical test?

A baseline claim of no effect or no difference, used for comparison. It assumes nothing special is happening, serving as the default position.
Find the point estimate (center) for the interval $[18,26]$ .

$22$ . The center is $\frac{18+26}{2} = 22$ .
Which option best describes a margin of error of $\pm 3\%$ for a proportion estimate?

The true proportion is likely within $3\%$ of the estimate. The confidence interval captures the true value with high probability.
What is the key condition needed to claim a treatment causes an effect?

A randomized experiment with random assignment to groups. Random assignment controls for confounding variables, isolating treatment effects.
What is the main risk of a voluntary response (self-selected) sample?

It tends to overrepresent strong opinions (bias). People with strong feelings are more likely to volunteer their responses.
Which conclusion is valid: A study finds $r=0.8$ between sleep and grades in one school; can you claim sleep causes higher grades?

No; correlation alone does not establish causation. Strong correlation doesn't prove one variable causes the other.
What is the key feature of a randomized experiment that reduces confounding?

Random assignment to treatment groups. Randomization balances both known and unknown variables across groups.
Which claim is justified: 'The drug caused improvement' based on an observational study with no random assignment?

Not justified; without random assignment, causation is not supported. Observational studies can show association but not causation.
Which statement is correct about correlation $r$ and causation?

A large $|r|$ does not prove causation. Correlation measures association strength, not causal relationships.
What is the key condition that allows a sample result to be generalized to a population?

A random sample from the target population. Random sampling ensures the sample represents the population without bias.
Identify the correct conclusion: a random sample of $200$ voters shows $54\%$ support for A.

Generalize to the population of voters, not claim A causes support. Random sampling allows generalization, not causation.
What is the correct interpretation if a margin of error is $\pm 3$ for an estimated mean of $50$ ?

A plausible range is $[47,53]$ . Add and subtract ME from the estimate to get the interval.
Which conclusion is supported if a $95\%$ confidence interval for $\mu_1-\mu_2$ is entirely negative?

There is evidence that $\mu_1<\mu_2$ . All negative values mean $\mu_1-\mu_2<0$ , so $\mu_1<\mu_2$ .
What is nonresponse bias in a survey?

Selected individuals do not respond, and responders differ from nonresponders. Missing responses bias results if non-responders have different characteristics.
What is the effect of a larger population size on margin of error for a fixed $n$ ?

Essentially no effect (if sampling fraction is small). For small sampling fractions, population size doesn't affect precision.
What is the key difference between a parameter and a statistic?

Parameter: population value; statistic: sample value. Parameters describe entire populations; statistics describe samples.
A $95\%$ confidence interval for a proportion is $[0.30,0.38]$ . Is $0.40$ plausible?

No, because $0.40$ is outside the interval. Values outside the confidence interval are considered implausible.
Which option is a valid reason a reported association might not be causal?

A confounding variable could explain the association. Third variables can create non-causal associations.
Which option is the best measure of spread for skewed data with outliers: $IQR$ or standard deviation?

$IQR$ . Based on quartiles, $IQR$ isn't affected by extreme outliers.
Identify the best conclusion if a confidence interval for a population mean excludes $0$ .

There is evidence the true mean is not $0$ . If $0$ is outside the interval, the mean significantly differs from $0$ .
Choose the correct conclusion: $52\%\pm 4\%$ support a policy; does this support a majority claim?

No; the interval includes $50\%$ (from $48\%$ to $56\%$ ). $50\%$ falls within the confidence interval, so majority is uncertain.
Which phrase correctly describes a correlation between two variables?

An association; it does not by itself prove causation. Correlation shows relationship but other factors could be the true cause.
What is the margin of error for an interval estimate written as $\text{estimate}\pm E$ ?

$E$ . The $\pm$ value represents the margin of error.
What is selection bias in a survey or study?

Bias from a nonrandom sample that overrepresents some groups. Occurs when sampling method favors certain groups over others.
What is the main advantage of a randomized experiment over an observational study?

Random assignment reduces confounding, supporting cause-and-effect claims. It ensures treatment groups are similar except for the treatment itself.
Which conclusion is valid if subjects are randomly assigned to treatment and control groups?

A causal conclusion about the treatment's effect is justified. Random assignment allows causal inference.
Identify the best conclusion: a $95\%$ interval for $p$ is $[0.42,0.50]$ ; does it support $p=0.40$ ?

No; $0.40$ is not in $[0.42,0.50]$ . Values outside the confidence interval are not supported.
What is the correct generic form of a confidence interval for a parameter?

$\text{estimate} \pm \text{margin of error}$ . The interval extends from estimate minus ME to estimate plus ME.
What happens to typical margin of error when sample size increases (all else equal)?

It decreases. Larger samples provide more precise estimates with smaller margins.
Choose the correct comparison: intervals $[12,18]$ and $[16,22]$ overlap; can you claim a clear difference?

No; overlap suggests no clear difference from intervals alone. Overlapping intervals indicate the difference may not be significant.
Choose the correct claim: A study randomly assigns students to $2$ study methods and compares scores.

A cause-and-effect claim is justified. Random assignment allows causal inference about method effectiveness.
What is the main reason a voluntary response sample is unreliable for inference?

It has self-selection bias, so it is often not representative. People who volunteer often have stronger opinions than the general population.
What is the point estimate for a population proportion $p$ based on a sample?

$\hat{p}=\frac{x}{n}$ . Sample proportion estimates population proportion.
Which study type can best justify a cause-and-effect claim: observational study or randomized experiment?

Randomized experiment. Only experiments with random assignment can establish causation.
What does a $95\%$ confidence interval for a population proportion mean?

The method captures the true proportion about $95\%$ of the time. In repeated sampling, $95\%$ of intervals contain the true value.
What is the main purpose of a randomized experiment compared with an observational study?

To support cause-and-effect conclusions. Random assignment isolates treatment effect from confounding variables.
What type of study can support association but not causation?

An observational study. Without manipulation, we can only observe relationships, not prove causation.
Identify the likely issue: a survey about school lunch is taken only during honors math class.

Sampling bias (undercoverage of non-honors students). Only surveying honors students misses other student perspectives.
What is sampling bias?

Systematic undercoverage that makes a sample unrepresentative. Occurs when certain groups are excluded or underrepresented.
What is the formula for population density if $P$ people live in area $A$ ?

$\text{density}=\frac{P}{A}$ . Number of people per unit area.
Identify the more appropriate center for skewed data with outliers: mean or median?

Median. Median resists the influence of extreme values better than mean.
What is the definition of a margin of error for an estimate such as a mean or proportion?

Maximum expected difference between estimate and true value. It's the largest likely error in the estimate.
Find the correct claim: A sample mean is $50$ with margin of error $4$ . What values are consistent with the estimate?

$46$ to $54$ . Margin of error gives range: $50 \pm 4 = [46, 54]$ .

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