Justifying Claims: Difference of Two Means - AP Statistics

Two fertilizers are compared for mean tomato yield (in pounds per plant). The claim is that Fertilizer X produces a greater mean yield than Fertilizer Y. A 90% confidence interval for $(\mu_X - \mu_Y)$ is $(0.0, 2.6)$ . Is the claim supported by the confidence interval?
Answer: No, because the interval contains 0, so it does not support the claim that $\mu_X > \mu_Y$ . This AP Statistics question examines justifying claims about the difference of two means with a confidence interval. The 90% interval for $\mu_X - \mu_Y$ is (0.0, 2.6) pounds, which includes zero, so it does not support the claim that Fertilizer X produces a greater mean yield than Y. Distractor choice C wrongly states that a non-negative interval proves $\mu_X > \mu_Y$ , but inclusion of zero allows for equality. Mini-lesson: Evidence for $\mu1 > \mu2$ requires the interval for $\mu1 - \mu2$ to be entirely above zero; including zero suggests no difference is plausible. Here, zero being in the interval means the claim lacks support. The interval's lower bound at exactly zero highlights the boundary case of no difference.
Two fertilizers are compared by measuring mean tomato yield (in pounds per plant). Fertilizer X had a sample mean yield of 6.4 and Fertilizer Y had a sample mean yield of 6.6. A 95% confidence interval for the difference in population means, defined as $\mu_X-\mu_Y$ , is $(-0.5, 0.2)$ . A gardener claims, "Fertilizer X produces a lower mean yield than Fertilizer Y." Is the claim supported by the confidence interval?
Answer: No, because the interval includes 0, so the data do not provide convincing evidence that $\mu_X<\mu_Y$ . This question tests whether a confidence interval supports a claim about population means when the interval includes 0. The interval (-0.5, 0.2) for μ_X - μ_Y contains both negative and positive values, including 0. When 0 is in the confidence interval, it means the true difference could be zero (no difference between population means). Therefore, we cannot conclude with confidence that μ_X < μ_Y. The key insight is that for a confidence interval to support a directional claim (like one mean being less than another), the entire interval must be on one side of 0. Since this interval straddles 0, the data do not provide convincing evidence for the gardener's claim.
A researcher claims that the mean reaction time is lower for people after drinking caffeinated tea than after drinking water. A 99% confidence interval for $(\mu_{\text{tea}}-\mu_{\text{water}})$ is $(-18,\ 5)$ milliseconds. Is the claim supported by this interval?
Answer: No, because the interval includes 0, so no difference is plausible. This question asks whether tea lowers reaction time compared to water. The interval (-18, 5) for (μ_tea - μ_water) contains 0, meaning we cannot rule out the possibility that the two means are equal at the 99% confidence level. Since 0 is a plausible value for the difference, we cannot conclude that μ_tea < μ_water. Choice E incorrectly interprets the presence of negative values as proof of the claim, but when an interval contains both positive and negative values, no directional conclusion is supported. For a confidence interval to support a claim that one mean is less than another, the entire interval must be negative (below 0).
A hospital compares mean length of stay (days) for patients receiving a new discharge protocol (New) versus the old protocol (Old). A 98% confidence interval for $\mu_{New}-\mu_{Old}$ is $(-1.9,\ -0.4)$ . The hospital claims the new protocol reduces mean length of stay. Is the claim supported by the interval?
Answer: Yes, because the entire interval is below 0, supporting that $\mu_{New}<\mu_{Old}$ . 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.
A pharmacist claims that a generic medication has the same mean time to symptom relief as the brand-name medication. A 95% confidence interval for $(\mu_{\text{generic}}-\mu_{\text{brand}})$ is $(-6,\ 3)$ minutes. Is the claim supported by this interval?
Answer: Yes, because 0 is in the interval, so equal means are plausible. 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.
A company tests two website designs and measures mean time-on-page. They compute a 95% confidence interval for the difference in population means, $\mu_A-\mu_B$ , as $(-12,\ 3)$ seconds. A manager claims: "Design A keeps users on the page longer on average than Design B." Is the claim supported by the confidence interval?
Answer: No; because the interval includes 0, the data do not support $\mu_A>\mu_B$ at the 95% level. 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.
Two teaching methods are compared using mean final-exam scores. A 95% confidence interval for the difference in population means, $\mu_{\text{Flipped}}-\mu_{\text{Lecture}}$ , is $( -1.5,\ 6.0 )$ points. A teacher claims: "Students in the flipped classroom score higher on average than students in the lecture classroom." Is the claim supported by the confidence interval?
Answer: No; because the interval contains 0, the data do not provide convincing evidence that $\mu_{\text{Flipped}} > \mu_{\text{Lecture}}$ . 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.
A psychologist compares mean reaction time (in milliseconds) for participants after drinking caffeinated coffee versus decaf. The claim is that caffeine leads to faster reactions on average. A 95% confidence interval for $(\mu_{\text{caffeine}}-\mu_{\text{decaf}})$ is $(-35,\ -5)$ . Is the claim supported by the confidence interval?
Answer: Yes, because the interval is entirely negative, supporting $\mu_{\text{caffeine}} < \mu_{\text{decaf}}$ (faster). In AP Statistics, this question assesses using confidence intervals to justify claims about two population means. The 95% interval for μ_caffeine - μ_decaf is (-35, -5) milliseconds, entirely negative, supporting that caffeine leads to faster (lower) mean reaction times. Choice A distracts by misinterpreting negative values as increasing time, confusing the claim's direction. For a claim of μ1 < μ2, the interval for μ1 - μ2 should be fully negative; otherwise, if it includes zero, evidence is lacking. The interval's position confirms the directional claim without implying causation for every individual. Confidence intervals provide ranges, not certainties about causes.
Two brands of batteries are tested for mean lifetime (hours). Brand A had a sample mean lifetime of 9.8 hours and Brand B had 10.1 hours. A 95% confidence interval for the difference in population means, defined as $\mu_A-\mu_B$ , is $(-0.9, -0.1)$ . A consumer claims, "Brand B lasts longer on average than Brand A." Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is below 0, which supports $\mu_A<\mu_B$ . 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.
A teacher compares mean quiz scores (percent) between a class that used practice quizzes (P) and a class that did not (N). A 95% confidence interval for $\mu_P-\mu_N$ is $(-4,\ 9)$ . The teacher claims using practice quizzes increases the mean quiz score. Is the claim supported by the interval?
Answer: No, because the interval includes 0, so it does not support that $\mu_P>\mu_N$ . 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.
A school compares mean nightly sleep for students who participate in a mindfulness program (M) versus those who do not (N). The researchers report a 95% confidence interval for the difference in population means, $\mu_M-\mu_N$ , as $(0.2,\ 1.1)$ hours. A student claims: "Students in the mindfulness program sleep more on average than students not in the program." Is the claim supported by the confidence interval?
Answer: Yes; because all values in the interval are positive, it supports $\mu_M>\mu_N$ . 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.
A city claims that households with smart thermostats use less electricity per month on average than households without smart thermostats. A 95% confidence interval for $(\mu_{\text{smart}}-\mu_{\text{not smart}})$ is $( -42,\ -10)$ kWh. Is the claim supported by this interval?
Answer: Yes, because the interval is entirely below 0, consistent with $\mu_{\text{smart}}<\mu_{\text{not smart}}$ . This question tests whether smart thermostats use less electricity on average. The interval (-42, -10) for (μ_smart - μ_not smart) is entirely negative, which means μ_smart < μ_not smart with 95% confidence. This supports the claim that households with smart thermostats use less electricity on average. Choice A incorrectly interprets negative values as meaning smart thermostats use more, when the opposite is true. When a confidence interval for (μ₁ - μ₂) is entirely negative, it provides evidence that the first population mean is smaller than the second. This interval structure directly supports claims about one group having a lower average than another.
A company compares mean battery life (hours) for two phone models, Model X and Model Y. A 95% confidence interval for $\mu_X-\mu_Y$ is reported as $(-1.6,\ 0.9)$ . The company claims Model X has longer mean battery life than Model Y. Is the claim supported by the interval?
Answer: No, because the interval contains 0, so it does not provide evidence that $\mu_X>\mu_Y$ . This question examines whether a confidence interval supports a comparative claim about means. The 95% confidence interval for μ_X - μ_Y is (-1.6, 0.9), which contains 0. When 0 is in the confidence interval for a difference of means, it indicates that zero difference is a plausible value, so we cannot conclude that one mean is larger than the other. The claim that Model X has longer battery life (μ_X > μ_Y) is not supported. Choice A incorrectly thinks having both positive and negative values supports the claim. Choice C misunderstands what 0 in the interval means. Choice D incorrectly interprets the negative part of the interval. Choice E wrongly treats the interval endpoints as exact values. Remember: if 0 is in the confidence interval for μ_1 - μ_2, we cannot conclude that μ_1 ≠ μ_2.
A company tests whether a new training program reduces mean time (in minutes) to complete a task. The claim is that trained employees are faster on average. A 90% confidence interval for $(\mu_{\text{trained}}-\mu_{\text{untrained}})$ is $(-4.8,\ -0.6)$ . Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is below 0, supporting $\mu_{\text{trained}} < \mu_{\text{untrained}}$ . 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.
A teacher compares mean quiz scores for students who used flashcards versus students who did not. The claim is that flashcard users score higher on average. A 95% confidence interval for $(\mu_{\text{flashcards}}-\mu_{\text{no flashcards}})$ is $(-1.2,\ 8.4)$ points. Is the claim supported by the confidence interval?
Answer: No, because the interval includes 0, so it does not provide convincing evidence that flashcards raise the mean score. The skill here in AP Statistics is justifying comparative claims about two means using a confidence interval. The 95% interval for μ_flashcards - μ_no flashcards is (-1.2, 8.4) points, including zero, so it lacks convincing evidence that flashcard users score higher on average. Choice A is a distractor, incorrectly focusing on the positive upper end without considering the inclusion of zero and negatives. For claims like μ1 > μ2, the interval for μ1 - μ2 must be entirely positive; inclusion of zero or negatives means the true difference could be zero or negative. This interval's range across zero indicates uncertainty in the direction of the effect. Confidence intervals estimate plausible values, not guarantees for individuals.
A company tests two advertising designs and records the mean number of website visits per day. Design 1 had a sample mean of 510 visits and Design 2 had a sample mean of 525 visits. A 90% confidence interval for the difference in population means, defined as $\mu_1-\mu_2$ , is $(-30, -5)$ . A manager claims, "Design 1 results in fewer mean daily visits than Design 2." Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is below 0, which supports $\mu_1<\mu_2$ . This question requires interpreting a confidence interval for the difference μ_1 - μ_2 = (-30, -5). Since the entire interval contains only negative values, we can be 90% confident that μ_1 - μ_2 < 0, which means μ_1 < μ_2. This directly supports the manager's claim that Design 1 results in fewer mean daily visits than Design 2. A common error is getting confused by the negative values, but negative values for μ_1 - μ_2 indicate that the first mean is smaller than the second. When a confidence interval for a difference is entirely below zero, it provides evidence that the first population mean is less than the second population mean.
A nutritionist compares mean LDL cholesterol for adults following Diet X versus Diet Y after 8 weeks. A 95% confidence interval for $\mu_X-\mu_Y$ is reported as $(-18,\ -4)$ mg/dL. The nutritionist claims: "Diet X results in a lower mean LDL cholesterol than Diet Y." Is the claim supported by the confidence interval?
Answer: Yes; because the entire interval is negative, it supports $\mu_X<\mu_Y$ . This question tests interpretation of a confidence interval where all values are negative. The 95% confidence interval for μ_X - μ_Y is (-18, -4) mg/dL, meaning every plausible value for the difference is negative. When μ_X - μ_Y < 0, this algebraically means μ_X < μ_Y, which supports the nutritionist's claim that Diet X results in lower mean LDL cholesterol than Diet Y. Students often confuse the direction: if the interval for μ₁ - μ₂ is entirely negative, then μ₁ < μ₂. Option D incorrectly interprets the interval as describing individual outcomes rather than population means. Option E incorrectly claims causation, which requires a randomized experiment, not just a confidence interval showing association.
A runner compares mean mile time after training with Plan P versus Plan Q. A 90% confidence interval for $\mu_P-\mu_Q$ is $(0.05,\ 0.40)$ minutes, where smaller times are faster. The runner claims: "Plan P produces faster (smaller) mean mile times than Plan Q." Is the claim supported by the confidence interval?
Answer: No; because the interval is positive, it suggests $\mu_P>\mu_Q$ , meaning Plan P is slower on average. This question involves interpreting a confidence interval when the context specifies that smaller values are better. The 90% confidence interval for μ_P - μ_Q is (0.05, 0.40) minutes, entirely positive. Since all values in the interval are positive, we can conclude μ_P > μ_Q, meaning Plan P produces larger (slower) mean mile times than Plan Q. The runner's claim that Plan P produces faster times is contradicted by the interval. When comparing means where smaller is better, a positive interval for μ₁ - μ₂ means the first group has worse (larger) outcomes. Students must carefully read whether larger or smaller values represent better outcomes. The confidence level (90% vs 95%) doesn't affect the interpretation of whether the interval supports a directional claim.
A professor claims that students in a flipped classroom format have a higher mean final exam score than students in a traditional format. A 95% confidence interval for $(\mu_{\text{flipped}}-\mu_{\text{traditional}})$ is $( -3,\ 9)$ points. Is the claim supported by this interval?
Answer: No, because the interval includes 0, so no difference is plausible. 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.
A researcher compares mean systolic blood pressure (mmHg) for adults who follow Diet A versus Diet B. The sample mean for Diet A was 121 and for Diet B was 125. A 99% confidence interval for the difference in population means, defined as $\mu_A-\mu_B$ , is $(-9, -1)$ . The researcher claims, "Diet A leads to a lower mean systolic blood pressure than Diet B." Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is below 0, supporting $\mu_A<\mu_B$ . This question involves interpreting a confidence interval that is entirely negative. The 99% confidence interval (-9, -1) for μ_A - μ_B contains only negative values, meaning we are 99% confident that μ_A - μ_B < 0. This is equivalent to saying μ_A < μ_B, which directly supports the researcher's claim that Diet A leads to lower mean systolic blood pressure than Diet B. Students often misinterpret what it means when an interval doesn't include 0 or misunderstand probability statements about confidence intervals. When comparing means, if the confidence interval for the difference is entirely below zero, it provides strong evidence that the first population mean is less than the second.

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