Making Inferences and Justifying Conclusions: Understanding Surveys, Experiments, and Observational Studies (CCSS.S-IC.3)
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Common Core High School Statistics And Probability › Making Inferences and Justifying Conclusions: Understanding Surveys, Experiments, and Observational Studies (CCSS.S-IC.3)
In a randomized experiment, 200 volunteers were randomly assigned to receive either a new reminder message or the standard message. In the new-message group, 68 of 100 responded; in the standard group, 54 of 100 responded (a 14-point difference). A randomization test (10,000 shuffles under no treatment effect) produced differences of 14 points or more in about 4% of trials. Which conclusion is supported by this simulation?
Because this was a survey, we cannot infer causality.
Random assignment plus a 4% tail probability provides evidence the new message improves response, though chance could still explain the observed difference.
We have proved the new message will increase response for every person.
Random sampling guarantees these results apply to all adults everywhere.
Explanation
Random assignment justifies a causal claim, and a 4% tail suggests the observed effect is unlikely by chance alone. It is evidence for a treatment effect, not proof, and does not guarantee universal or population-wide generalization without random sampling.
A random sample of 100 students found that 62 prefer adding a smoothie bar to the cafeteria. A simulation assuming $p=0.50$ (10,000 samples of size 100) produced 62 or more in about 7% of trials. Which conclusion is supported by this simulation?
Because this was an experiment, we can conclude the smoothie bar causes preference.
The sample proves a majority of all students prefer the smoothie bar.
The population proportion must equal 0.62 with 93% confidence.
The result is compatible with $p=0.50$, but it also suggests the true proportion could be above 0.50; more data would strengthen the claim.
Explanation
A 7% tail means the sample is plausible under $p=0.50$ yet leans toward a majority. Random sampling supports inference to the population, but this is not conclusive proof or an exact population proportion.
A die is rolled 60 times and 18 sixes are observed. A simulation of 10,000 sets of 60 fair rolls showed 18 or more sixes in 460 runs. Which conclusion is supported by this simulation?
Seeing 18 or more sixes is somewhat rare under a fair die, providing some evidence the die may roll sixes more often than fair.
The die's probability of six is 0.30 because 18 out of 60 were sixes.
The die is proven to be loaded.
This tells us nothing because dice rolls are random.
Explanation
About 4.6% of fair-die simulations had 18+ sixes, which is somewhat unlikely but possible by chance. This gives some evidence of a higher-than-fair six rate without proving it.
A student flips a coin 100 times and gets 62 heads. To check if this could happen by chance with a fair coin, a simulation of 1,000 runs of 100 fair flips found 23 runs with 60% or more heads. Which conclusion is supported by this simulation?
The coin is definitely biased toward heads.
The result is somewhat unlikely under a fair coin, providing some evidence the coin may be biased toward heads.
The coin's head probability is $p=0.62$.
Because the flips were random, no inference about fairness is possible.
Explanation
About 2.3% of simulated fair coins produced at least 60% heads, so the observed 62% is somewhat unlikely under $p=0.5$. That is evidence against fairness, but random variation means it is not proof.
A coin is flipped 50 times and lands heads 38 times. A fair-coin simulation (10,000 runs) yielded 38 or more heads in about 0.1% of trials. What does the result suggest about the population proportion?
The coin is fair; 38 heads out of 50 is exactly what we expect.
We can state with 99.9% certainty that the population proportion of heads is exactly 0.76.
The data provide strong evidence that $p$, the probability of heads, is greater than 0.5, though no simulation can prove it with absolute certainty.
Randomness makes inference impossible here.
Explanation
A 0.1% tail under the fair model is very unlikely, giving strong evidence that $p>0.5$. However, simulations support probabilistic conclusions and do not establish exact population proportions or absolute certainty.
A coin is flipped 50 times and lands tails 38 times. A simulation of 10,000 runs of 50 fair flips produced 38 or more tails in 2 runs. Which conclusion is supported by this simulation?
The coin's tail probability is $p=0.76$.
Getting 38 tails is extremely unlikely for a fair coin, so there is very strong evidence the coin is biased toward tails.
Because flips are random, no conclusion can be drawn about $p$.
The coin is unquestionably biased; there is zero chance a fair coin produced this result.
Explanation
About 0.02% of fair-coin simulations were as extreme, so the outcome is extraordinarily rare under $p=0.5$. This is very strong evidence of bias, but randomness prevents absolute certainty.
In an experiment, 200 users were randomly assigned: 100 received a reminder email and 100 did not. Sign-ups were 64 with the email and 52 without. A randomization test (shuffling labels) produced a difference of at least 12 percentage points in 7.8% of 10,000 trials. Which conclusion is supported by this simulation?
The reminder definitely causes higher sign-ups.
The reminder has no effect; 7.8% is too small to matter.
The population difference in sign-up rates is exactly 12%.
The observed difference would be somewhat uncommon by chance alone, so there is some evidence the reminder increases sign-ups; random assignment supports a causal explanation, but it is not certain.
Explanation
A 7.8% randomization p-value indicates some evidence against no effect. Because of random assignment (experiment), a causal interpretation is supported, but chance variation means the effect is not proven.
A student flips a coin 200 times and gets 118 heads. They simulate 10,000 sets of 200 fair-coin flips and see 118 or more heads about 9% of the time. Which conclusion is supported by this simulation?
The result is somewhat unusual but still plausible if the coin is fair; there is not strong evidence the coin is biased toward heads.
The coin is definitely biased toward heads because 9% is too small to be chance.
The coin is definitely fair because 118 is close to half.
The population proportion of heads is exactly 0.59.
Explanation
About 9% of fair-coin simulations match or exceed the observed result, so the outcome can occur by chance. This provides little evidence of bias and does not prove fairness or a specific population proportion.
In a random sample of 80 students, 54 prefer the new lunch menu. To test whether half the school prefers it, a simulation drew 10,000 random samples of size 80 from a population with $p=0.5$ and found 120 samples with 54 or more preferring. What does the result suggest about the population proportion?
The sample proportion 0.675 must equal the population proportion.
Because the sample was random, the result is meaningless.
The simulation suggests the true proportion $p$ is greater than 0.5.
The school definitely has more than half of all students preferring the new lunch.
Explanation
Only 1.2% of samples at $p=0.5$ were as extreme as the observed result, so there is evidence $p>0.5$. Random sampling justifies inferring about the population, but the conclusion is not certain.
In 180 rolls of a die, 40 were sixes. A simulation of 10,000 sets of 180 fair rolls produced 40 or more sixes in about 2% of trials. What does the result suggest about the population proportion?
The true proportion of sixes is exactly 22%.
Randomness plays no role; the die must be fair.
There is evidence that the probability of rolling a six may be greater than 1/6, but chance alone could still produce this outcome.
We can conclude with 98% certainty that the chance of a six is 1/4.
Explanation
A 2% tail under the fair model suggests the observed result is unlikely by chance, giving evidence the true probability could exceed 1/6, but it does not prove a specific value or eliminate chance entirely.