AP Statistics Quiz: Justifying Claims Difference Of Two Means

What this quiz covers

This quiz focuses on Justifying Claims Difference Of Two Means, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Yes, because the entire interval is below 0, supporting that $\mu_{New}<\mu_{Old}$. (correct answer)
2. No, because a negative interval means the new protocol increases length of stay.
3. No, because 0 is not in the interval, so there is no difference.
4. Yes, because 98% of patients under the new protocol stay fewer days than 98% under the old protocol.
5. Yes, because the interval proves the new protocol causes every patient to leave 0.4 to 1.9 days earlier.

Explanation: This question tests understanding of negative confidence intervals in context. The 98% confidence interval for μ_New - μ_Old is (-1.9, -0.4), which is entirely below 0. This means μ_New - μ_Old < 0, so μ_New < μ_Old, indicating that the new protocol has a lower mean length of stay. Since lower length of stay means patients leave sooner, the hospital's claim that the new protocol reduces mean length of stay is supported. Choice B incorrectly interprets negative values as increasing length of stay. Choice C misunderstands what 0 not being in the interval means. Choice D wrongly interprets the interval as describing individual patient percentiles. Choice E claims the interval applies to every patient and implies causation beyond what's justified. When μ_1 - μ_2 is entirely negative, it supports that μ_1 < μ_2.

Question 2

1. Yes, because the upper end of the interval is positive, so practice quizzes increase scores.
2. No, because the interval includes 0, so it does not support that $\mu_P>\mu_N$. (correct answer)
3. Yes, because including 0 means there is definitely some improvement.
4. No, because the interval is for $\mu_N-\mu_P$ and it crosses 0.
5. Yes, because the interval proves practice quizzes cause an increase between 4 and 9 points.

Explanation: This question examines a confidence interval that contains 0. The 95% confidence interval for μ_P - μ_N is (-4, 9), which includes 0. Since 0 is in the interval, we cannot conclude that the mean quiz scores differ between the two groups. The teacher's claim that practice quizzes increase the mean score (μ_P > μ_N) is not supported because 0 is a plausible value for the true difference. Choice A incorrectly focuses only on the positive end. Choice C misinterprets 0 being in the interval as confirming improvement. Choice D is partially correct about 0 being in the interval but incorrectly states the order. Choice E wrongly claims causation and misinterprets the interval. When 0 is contained in a confidence interval for μ_1 - μ_2, we cannot make comparative claims about the means.

Question 3

1. No; because the interval contains 0, there may be no difference in mean sleep.
2. Yes; because all values in the interval are positive, it supports $\mu_M>\mu_N$. (correct answer)
3. No; because the interval is for $\mu_N-\mu_M$, not $\mu_M-\mu_N$.
4. Yes; because 95% of students in the program sleep more than those not in the program.
5. Yes; because the interval proves the program causes more sleep.

Explanation: This question tests understanding of how to interpret confidence intervals when comparing two population means. The 95% confidence interval for μ_M - μ_N is (0.2, 1.1) hours, where M represents students in the mindfulness program and N represents those not in the program. Since the entire interval contains only positive values, we can conclude with 95% confidence that μ_M > μ_N, meaning students in the mindfulness program sleep more on average. The key insight is that when a confidence interval for μ₁ - μ₂ contains only positive values, it supports the claim that μ₁ > μ₂. Option D incorrectly interprets the interval as describing individual students rather than population means, while option E incorrectly claims causation when the interval only shows association.

Question 4

1. Yes; because the interval contains some values above 0, the flipped method is higher.
2. No; because the interval contains 0, the data do not provide convincing evidence that $\mu_{\text{Flipped}} > \mu_{\text{Lecture}}$. (correct answer)
3. Yes; because the interval contains 0, it confirms no difference and therefore supports the claim.
4. No; because the interval is for $\mu_{\text{Lecture}}-\mu_{\text{Flipped}}$, not the stated order.
5. Yes; because 95% of flipped-class students score higher than lecture students.

Explanation: This question addresses a common scenario where a confidence interval contains zero. The 95% confidence interval for μ_Flipped - μ_Lecture is (-1.5, 6.0) points, which includes negative values, zero, and positive values. Because zero is within the interval, it's plausible that μ_Flipped = μ_Lecture, meaning the data do not provide convincing evidence at the 95% confidence level that the flipped classroom produces higher mean scores. The teacher's claim requires μ_Flipped > μ_Lecture, which would need the entire interval to be positive. Option A incorrectly focuses only on the positive values while ignoring that the interval also contains negative values and zero. When interpreting confidence intervals for differences, the key question is whether zero is included—if it is, we cannot conclude there's a difference between the means.

Question 5

1. No, because the interval is small, so the difference is not real.
2. Yes, because the entire interval is above 0, supporting that $\mu_S>\mu_N$. (correct answer)
3. No, because 0 is not in the interval, so there is no difference.
4. Yes, because 90% of students who attend have higher GPAs than 90% who do not.
5. Yes, because the interval proves attending causes each student's GPA to increase by 0.05 to 0.30.

Explanation: This question tests interpretation of a positive confidence interval. The 90% confidence interval for μ_S - μ_N is (0.05, 0.30), which is entirely above 0. This means μ_S - μ_N > 0, so μ_S > μ_N, supporting the claim that students who attended supplemental instruction have a higher mean GPA. Choice A incorrectly dismisses the interval based on its width. Choice C misinterprets 0 not being in the interval as meaning no difference exists. Choice D wrongly interprets the interval as describing individual student percentiles. Choice E incorrectly claims causation and misinterprets the interval as applying to individual changes. When a confidence interval for μ_1 - μ_2 is entirely positive, it provides evidence that μ_1 > μ_2 at the stated confidence level.

Question 6

1. Yes; because the entire interval is negative, it supports $\mu_G<\mu_N$. (correct answer)
2. No; because 99% confidence means there is a 99% chance the true difference is 0.
3. No; because the interval is negative, it indicates the group has higher stress.
4. Yes; because the interval guarantees the support group causes lower stress.
5. No; because the interval includes 0, so no conclusion can be made.

Explanation: This question involves interpreting a negative confidence interval where lower scores are better. The 99% confidence interval for μ_G - μ_N is (-5.0, -0.5), entirely negative, where G represents the support group and N represents not in the group. Since higher scores mean more stress and the interval is entirely negative, we can conclude μ_G < μ_N, meaning students in the support group have lower mean stress. The counselor's claim is supported. Option B misunderstands what confidence level means—99% confidence refers to the reliability of the interval estimation method, not the probability that the true difference is zero. Option C incorrectly interprets negative values as indicating higher stress for the group. When interpreting intervals, always consider both the direction of the difference and what constitutes a desirable outcome.

Question 7

1. Yes, because the entire interval is below 0, which supports $\mu_A<\mu_B$. (correct answer)
2. No, because the interval does not include 0, so the brands must be the same.
3. No, because the confidence interval is for $\mu_B-\mu_A$ rather than $\mu_A-\mu_B$.
4. Yes, because the sample means differ by 0.3 hours, so Brand B must be longer in the population.
5. Yes, because 95% confidence means 95% of Brand B batteries last longer than Brand A batteries.

Explanation: This question involves interpreting a negative confidence interval to support a claim about which mean is larger. The interval (-0.9, -0.1) for μ_A - μ_B is entirely negative, meaning we are 95% confident that μ_A - μ_B < 0. This is equivalent to μ_A < μ_B, which supports the consumer's claim that Brand B lasts longer on average than Brand A. Students sometimes get confused about the order of subtraction or misinterpret confidence levels as applying to individual items rather than the parameter. When comparing two means, if the confidence interval for their difference is entirely negative, it provides evidence that the first mean is less than the second mean.

Question 8

1. Yes, because the interval includes positive values, so flipped must be higher.
2. No, because the interval includes 0, so no difference is plausible. (correct answer)
3. Yes, because the interval's midpoint is positive, so the claim is proven.
4. No, because the interval is for $(\mu_{\text{traditional}}-\mu_{\text{flipped}})$, so it actually supports the claim.
5. Yes, because 95% confidence means there is a 95% chance the flipped mean is higher.

Explanation: This question asks whether flipped classrooms produce higher exam scores. The interval (-3, 9) for (μ_flipped - μ_traditional) contains 0, meaning a difference of 0 is plausible at the 95% confidence level. Since we cannot rule out the possibility that the two teaching formats produce equal mean scores, the claim that μ_flipped > μ_traditional is not supported. Choice C incorrectly focuses on the positive midpoint rather than the fact that 0 is included. When a confidence interval contains both negative and positive values, it indicates uncertainty about the direction of the difference, and no directional claim can be supported.

Question 9

1. Yes, because 0 is in the interval, so equal means are plausible. (correct answer)
2. No, because the interval includes negative values, so the generic is definitely faster.
3. No, because the interval includes positive values, so the brand is definitely faster.
4. Yes, because the interval proves the two means are exactly equal.
5. No, because to support equality the interval must be centered at 0.

Explanation: This question tests understanding of claims about equality of means. The interval (-6, 3) for (μ_generic - μ_brand) contains 0, which means a difference of 0 (equal means) is plausible at the 95% confidence level. This supports the claim that the two medications have the same mean time to relief. Choices B and C incorrectly interpret the presence of negative or positive values as definitive evidence against equality. When testing a claim of equality between two means, we look for whether 0 is contained in the confidence interval. If 0 is included, the claim of equal means is supported; if 0 is excluded, the claim is not supported.

Question 10

1. Yes; because the interval includes positive values, A must be better.
2. No; because the interval includes 0, the data do not support $\mu_A>\mu_B$ at the 95% level. (correct answer)
3. Yes; because the entire interval is below 0, so $\mu_A>\mu_B$.
4. No; because a 95% confidence interval means there is a 95% chance that $\mu_A=\mu_B$.
5. Yes; because the interval proves A causes longer time-on-page.

Explanation: This question examines whether a confidence interval supports a directional claim about two means. The 95% confidence interval for μ_A - μ_B is (-12, 3) seconds, which includes both negative and positive values, crucially including zero. When a confidence interval for a difference contains zero, it means the data do not provide convincing evidence of a difference between the population means at the stated confidence level. Since zero is a plausible value for μ_A - μ_B, we cannot conclude that μ_A > μ_B. The manager's claim that Design A keeps users on the page longer is not supported because the interval suggests μ_A could be less than, equal to, or greater than μ_B. Remember: for a confidence interval to support μ₁ > μ₂, the entire interval for μ₁ - μ₂ must be positive.

Question 11

1. Yes, because the interval is positive, so $\mu_{P1}>\mu_{P2}$.
2. No, because the interval is for $\mu_{P2}-\mu_{P1}$ and it is entirely above 0, supporting $\mu_{P2}>\mu_{P1}$. (correct answer)
3. Yes, because 0 is not in the interval, so Program 1 must be better.
4. No, because a 90% interval is never enough to support any comparative claim.
5. Yes, because the interval proves Program 2 causes 1.0 to 5.4 more points for every student.

Explanation: This question tests understanding of the order of subtraction in confidence intervals. The 90% confidence interval is for μ_P2 - μ_P1 and equals (1.0, 5.4). Since this interval is entirely positive, it means μ_P2 - μ_P1 > 0, which implies μ_P2 > μ_P1. Therefore, Program 2 has a higher mean improvement than Program 1, not the other way around. The student's claim that Program 1 leads to higher mean improvement is not supported. Choice A incorrectly ignores the order of subtraction. Choice C misinterprets what 0 not being in the interval means. Choice D incorrectly dismisses 90% confidence intervals. Choice E wrongly claims causation and misinterprets the interval as applying to individuals. When interpreting confidence intervals for differences, always pay attention to which mean is subtracted from which.

Question 12

1. Yes, because part of the interval is below 0, so coffee is definitely faster.
2. No, because the interval includes 0, so it does not support a difference in mean reaction time. (correct answer)
3. Yes, because 0 is in the interval, which confirms coffee has an effect.
4. No, because the interval shows $\mu_W-\mu_C$ is negative, so coffee is slower.
5. Yes, because the interval proves coffee reduces reaction time by exactly 35 ms.

Explanation: This question examines a confidence interval that contains 0. The 95% confidence interval for μ_C - μ_W is (-35, 5), which includes 0. When 0 is in the confidence interval for a difference of means, we cannot conclude that the means differ. The researcher claims coffee leads to faster (smaller) reaction times, which would require μ_C < μ_W, but the interval doesn't support this because 0 is a plausible value for the difference. Choice A incorrectly focuses only on part of the interval. Choice C misinterprets 0 being in the interval as confirming an effect. Choice D misreads the interval. Choice E wrongly treats the interval as an exact value and claims causation. Remember: containing 0 means we cannot reject the null hypothesis of no difference between means.

Question 13

1. Yes, because the entire interval is below 0, supporting that $\mu_M<\mu_N$. (correct answer)
2. No, because the interval is negative, which means meditation increases heart rate.
3. No, because 0 is not in the interval, so there is no association.
4. Yes, because 95% of meditators have lower heart rates than 95% of non-meditators.
5. Yes, because the interval proves meditation causes a decrease of between 1.2 and 6.5 bpm.

Explanation: This question involves interpreting a negative confidence interval for a difference of means. The 95% confidence interval for μ_M - μ_N is (-6.5, -1.2), which is entirely below 0. This means μ_M - μ_N < 0, so μ_M < μ_N, indicating that meditators have a lower mean resting heart rate than non-meditators. The claim is supported. Choice B incorrectly interprets negative values as meaning meditation increases heart rate. Choice C misunderstands what 0 not being in the interval signifies. Choice D wrongly interprets the interval as describing individual percentiles rather than population means. Choice E incorrectly claims causation from what may be an observational study. When a confidence interval for μ_1 - μ_2 is entirely negative, it supports the claim that μ_1 < μ_2.

Question 14

1. No, because the interval contains negative values, so trained employees must be slower.
2. Yes, because the entire interval is below 0, supporting $\mu_{\text{trained}} < \mu_{\text{untrained}}$. (correct answer)
3. No, because 0 is not included, so we cannot make any comparison.
4. Yes, because the interval does not include 0, which proves the training causes faster times.
5. No, because a 90% interval is not reliable enough to support any claim.

Explanation: This question tests the ability to justify claims about the difference of two means using a confidence interval in AP Statistics. The 90% confidence interval for μ_trained - μ_untrained is (-4.8, -0.6) minutes, entirely below zero, supporting the claim that trained employees have a lower mean time (faster) than untrained ones. A frequent distractor is choice A, which incorrectly assumes negative values mean trained are slower, ignoring the claim's direction. For comparative claims, if the interval for μ1 - μ2 is entirely negative, it suggests μ1 < μ2; including zero means no evidence of a difference. In this case, the interval's position entirely below zero aligns with the claim of faster times for trained employees. Note that confidence level affects interval width but not the interpretation relative to zero.

Question 15

1. Yes, because the entire interval is below 0, supporting $\mu_A<\mu_B$ (faster time). (correct answer)
2. No, because negative values mean Drug A is slower than Drug B.
3. No, because 0 is not in the interval, so there is no evidence of a difference.
4. Yes, because 95% confidence means Drug A will be faster for 95% of patients.
5. No, because the interval should be interpreted as $\mu_B-\mu_A$, which would be positive.

Explanation: This question involves interpreting a negative confidence interval in the context of time measurements where lower is better. The interval (-8, -1) for μ_A - μ_B is entirely negative, meaning μ_A - μ_B < 0, or equivalently, μ_A < μ_B. Since we're measuring time until relief (where lower values are better), having μ_A < μ_B means Drug A provides relief faster on average than Drug B. This directly supports the doctor's claim. Students often get confused when "smaller is better" scenarios arise, but the interpretation remains consistent: a negative difference means the first value is smaller than the second. When a confidence interval is entirely negative, it provides evidence that the first population mean is less than the second.

Question 16

1. No, because the interval is positive, which means drivers commute longer on average.
2. Yes, because the interval contains 0, so transit is longer.
3. Yes, because the entire interval is above 0, supporting $\mu_{transit}>\mu_{drive}$. (correct answer)
4. No, because a 90% confidence interval is not high enough confidence to support any claim.
5. Yes, because 90% confidence means there is a 90% chance the true difference is between 2 and 10 minutes.

Explanation: This question tests interpretation of a confidence interval that is entirely positive. The 90% confidence interval (2, 10) for μ_transit - μ_drive contains only positive values, meaning we are 90% confident that μ_transit - μ_drive > 0. This translates to μ_transit > μ_drive, which directly supports the planner's claim that public transit users have a longer mean commute time than drivers. A common misconception is about what confidence level means - it's not about individual probabilities but about the reliability of the interval estimation method. When a confidence interval for a difference is entirely above zero, it provides evidence that the first population mean is greater than the second.

Question 17

1. Yes; because the interval is positive, it supports Brand S lasting longer.
2. No; because the interval for $\mu_R-\mu_S$ being positive supports Brand R lasting longer, not Brand S. (correct answer)
3. Yes; because the interval does not include 0, both brands must last longer than the other.
4. No; because any confidence interval comparing brands must include 0 to be valid.
5. Yes; because 95% confidence means Brand S will last longer in 95% of batteries.

Explanation: This question tests understanding of how the order of subtraction affects interpretation. The confidence interval (2.5, 7.0) hours is given for μ_R - μ_S, which is entirely positive, indicating μ_R > μ_S (Brand R lasts longer on average). However, the consumer misreads this as an interval for μ_S - μ_R. If the interval for μ_R - μ_S is positive, then the interval for μ_S - μ_R would be negative (specifically (-7.0, -2.5)), which would indicate μ_S < μ_R. Therefore, the consumer's claim that Brand S lasts longer is not supported—in fact, the interval supports the opposite conclusion. This highlights the critical importance of paying attention to the order of subtraction when interpreting confidence intervals for differences. The actual data support Brand R lasting longer, not Brand S.

Question 18

1. No; because the interval is positive, it shows Treatment A has a shorter stay.
2. Yes; because the entire interval is above 0, it supports $\mu_A>\mu_B$. (correct answer)
3. No; because confidence intervals cannot be used to compare means.
4. Yes; because 95% of patients on Treatment A stay longer than patients on Treatment B.
5. Yes; because the interval proves Treatment A causes longer stays.

Explanation: This question tests straightforward interpretation of a positive confidence interval. The 95% confidence interval for μ_A - μ_B is (1.2, 3.8) days, entirely positive. Since all values in the interval are greater than zero, we can conclude with 95% confidence that μ_A > μ_B, meaning patients receiving Treatment A stay longer in the hospital on average than those receiving Treatment B. The doctor's claim is directly supported by the interval. Option A incorrectly reverses the interpretation of positive values. Option D misinterprets the interval as describing individual patient outcomes rather than population means. When a confidence interval for μ₁ - μ₂ contains only positive values, it provides convincing evidence that μ₁ > μ₂ at the stated confidence level.

Question 19

1. Yes; because the interval is mostly negative, coffee is faster.
2. No; because the interval includes 0, it does not provide convincing evidence that $\mu_C<\mu_T$. (correct answer)
3. Yes; because the interval includes 0, it confirms coffee is faster.
4. No; because the interval being near 0 means the true difference must be exactly 0.
5. Yes; because a 95% confidence interval guarantees coffee is faster for 95% of people.

Explanation: This question examines a confidence interval that contains zero. The 95% confidence interval for μ_C - μ_T is (-0.08, 0.01) seconds, which includes both negative and positive values, including zero. Since zero is a plausible value for the difference, the data do not provide convincing evidence that coffee leads to faster mean reaction times than tea. The researcher's claim requires μ_C < μ_T (since smaller times are faster), which would need the entire interval to be negative. While most of the interval is negative, the inclusion of positive values and zero means we cannot conclude at the 95% confidence level that there's a difference. Option A incorrectly focuses on the interval being "mostly negative" while ignoring that it must be entirely negative to support the claim.

Question 20

1. No; because the interval is negative, it indicates solar homes use more electricity.
2. Yes; because all values are negative, it supports $\mu_S<\mu_N$. (correct answer)
3. No; because the interval includes 0, there may be no difference.
4. Yes; because 95% of solar homes use less electricity than 95% of non-solar homes.
5. Yes; because the interval proves installing solar panels causes lower grid usage.

Explanation: This question tests understanding of negative confidence intervals. The 95% confidence interval for μ_S - μ_N is (-250, -60) kWh, where S represents homes with solar panels and N represents homes without. Since every value in the interval is negative, we can conclude with 95% confidence that μ_S < μ_N, meaning homes with solar panels use less electricity from the grid on average. The resident's claim is supported because the entire interval being negative provides convincing evidence of the direction of the difference. Option A incorrectly interprets negative values as indicating higher usage. Option D misinterprets the interval as describing individual homes rather than population means. Remember: confidence intervals describe plausible values for population parameters, not individual outcomes.