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Sampling for Differences in Sample Proportions - AP Statistics

  1. A university compares the proportion of students who report being satisfied with dining services between on-campus and off-campus students. An SRS of n1=90n_1=90 on-campus students and an independent SRS of n2=110n_2=110 off-campus students are surveyed; p^1\hat p_1 and p^2\hat p_2 are the sample proportions satisfied. The sampling distribution of p^1−p^2\hat p_1-\hat p_2 is modeled for repeated sampling. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat p_1-\hat p_2 describes how p^1−p^2\hat p_1-\hat p_2 varies from sample to sample. This question distinguishes between what varies in a sampling distribution. The sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 describes how the sample statistic p^1−p^2\hat{p}_1 - \hat{p}_2 varies from sample to sample when we repeat the sampling process. The population parameters p1p_1 and p2p_2 are fixed and don't vary (eliminating A). It's not about individual responses but about the difference in proportions (eliminating C). The distributions of p^1\hat{p}_1 alone and p^1−p^2\hat{p}_1 - \hat{p}_2 are different (eliminating D). The center is p1−p2p_1 - p_2 regardless of whether sample sizes are equal (eliminating E).
  2. A tech company compares the proportion of users who click a new button design on two versions of an app. Version A is shown to an independent random sample of n1=250n_1=250 users and Version B to n2=250n_2=250 users; p^1\hat p_1 and p^2\hat p_2 are the sample click proportions. The company considers the sampling distribution of p^1−p^2\hat p_1-\hat p_2 over many repetitions. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat p_1-\hat p_2 is centered at p1−p2p_1-p_2, and its spread depends on p1p_1, p2p_2, n1n_1, and n2n_2. This question addresses both center and spread of the sampling distribution. The sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 is centered at the population difference p1−p2p_1 - p_2, and its spread (standard deviation) depends on all four values: p1p_1, p2p_2, n1n_1, and n2n_2 through the formula p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}. Even if p1−p2>0p_1 - p_2 > 0, sampling variability means some samples could yield negative differences (eliminating A). The center is not the sample statistic (eliminating C). The distribution describes sample statistics, not populations (eliminating D). Population sizes aren't relevant to the sampling distribution (eliminating E).
  3. A streaming service tests whether the proportion of users who finish a new series differs between two age groups. An independent random sample of n1=150n_1=150 users ages 18–34 and n2=150n_2=150 users ages 35+ is selected, and p^1\hat p_1 and p^2\hat p_2 are the sample proportions who finish the series. Consider the sampling distribution of p^1−p^2\hat p_1-\hat p_2 across repeated sampling. Which statement is correct?
    Answer: If the repeated-sampling conditions are met, p^1−p^2\hat p_1-\hat p_2 is approximately normal and centered at p1−p2p_1-p_2. This question examines conditions for the sampling distribution of differences. When proper conditions are met (random sampling, independence, and success-failure conditions), the sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 is approximately normal and centered at p1−p2p_1 - p_2. Equal sample sizes don't make the mean 0 unless p1=p2p_1 = p_2 (eliminating A). The distribution can be approximately normal despite proportions being bounded (eliminating C). The standard deviation formula is p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}, not the given expression (eliminating D). Random sampling is always necessary for valid inference (eliminating E).
  4. A streaming service compares the proportion of users who finish a new series in two regions. Region R: an SRS of nR=500n_R=500 users; Region S: an independent SRS of nS=50n_S=50 users. The statistic is p^R−p^S\hat{p}_R-\hat{p}_S. In repeated sampling, the sampling distribution is centered at pR−pSp_R-p_S and its standard deviation is influenced by both sample sizes. Which statement is correct?
    Answer: The sampling distribution's variability is driven more by the smaller sample size nSn_S than by the larger nRn_R. This question examines how unequal sample sizes influence the sampling distribution of proportion differences, a critical AP Statistics concept for real-world studies with varying group sizes. The spread is more affected by the smaller sample ((nSn_S = 50)) because its term in the standard deviation formula contributes more variance relative to the larger (nRn_R = 500). Choice A distracts by claiming more variability due to the large sample, which is backward since larger samples reduce individual variance contributions. Mini-lesson: in difference distributions from independent samples, the total spread combines variances additively, with smaller samples driving more uncertainty, while the center remains at (pRp_R - pSp_S). Unequal sizes don't prevent normality if conditions are met. This highlights the importance of bolstering smaller groups for balanced precision.
  5. A public health researcher compares vaccination rates in two cities. From City A, an SRS of n1=200n_1=200 adults is selected; from City B, an independent SRS of n2=50n_2=50 adults is selected. Let p^1\hat p_1 and p^2\hat p_2 be the sample proportions vaccinated, and consider the sampling distribution of p^1−p^2\hat p_1-\hat p_2 over many repetitions. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat p_1-\hat p_2 is centered at p1−p2p_1-p_2. This question tests understanding of center and spread in sampling distributions for differences. The sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 is centered at the true population difference p1−p2p_1 - p_2, regardless of sample sizes. Even with large samples, there is still sampling variability (eliminating A). The center is not determined by the first sample's result (eliminating C). The standard deviation depends on both the sample sizes AND the population proportions through the formula p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} (eliminating D). The distribution can be approximately normal with unequal sample sizes if conditions are met (eliminating E).
  6. A political analyst compares the proportion of voters who favor Candidate X in two counties. An independent random sample of n1=500n_1=500 registered voters from County 1 and n2=500n_2=500 from County 2 is taken; p^1\hat p_1 and p^2\hat p_2 are the sample proportions favoring Candidate X. Over repeated sampling, consider the sampling distribution of p^1−p^2\hat p_1-\hat p_2. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat p_1-\hat p_2 is approximately normal if the success–failure condition is met in both groups. This question tests understanding of normality conditions for sampling distributions. The sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 is approximately normal when the success-failure condition is met in both groups (typically np≥10np \geq 10 and n(1−p)≥10n(1-p) \geq 10 for each group). Large samples don't eliminate all variability (eliminating A). Normality doesn't require proportions near 0.5 (eliminating C). The population distributions don't need to be normal for the sampling distribution to be approximately normal (eliminating D). There is still spread due to sampling variability regardless of sample size (eliminating E).
  7. A company compares the proportion of customers who would recommend its service in two regions. An independent random sample of n1=60n_1=60 customers from Region 1 and n2=60n_2=60 customers from Region 2 is taken; the sample proportions who would recommend are p^1\hat p_1 and p^2\hat p_2. Over many repetitions, the sampling distribution of p^1−p^2\hat p_1-\hat p_2 is considered. Which statement is correct?
    Answer: If p1=p2p_1=p_2, then the sampling distribution of p^1−p^2\hat p_1-\hat p_2 is centered at 0. This question focuses on properties of the sampling distribution when comparing two proportions. When the true population proportions are equal (p1=p2p_1 = p_2), the difference p1−p2=0p_1 - p_2 = 0, so the sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 is centered at 0. However, individual samples will still show variation due to sampling variability, so p^1−p^2\hat{p}_1 - \hat{p}_2 won't equal 0 in every sample (eliminating B). The center is always p1−p2p_1 - p_2, not the observed difference (eliminating C). The distribution is approximately normal under proper conditions, not uniform (eliminating D). Changing sample sizes does affect the spread through the standard error formula (eliminating E).
  8. A public health researcher wants to compare the proportion of adults who got a flu shot this year in two cities. A simple random sample of n1=200n_1=200 adults is taken from City A and n2=200n_2=200 adults is taken from City B, and the statistic p^A−p^B\hat{p}_A-\hat{p}_B is computed. If these samples were repeatedly taken the same way, the sampling distribution of p^A−p^B\hat{p}_A-\hat{p}_B would be approximately normal and centered at pA−pBp_A-p_B, with a standard deviation that depends on pAp_A, pBp_B, n1n_1, and n2n_2. Which statement is correct?
    Answer: The sampling distribution of p^A−p^B\hat{p}_A-\hat{p}_B is centered at the true difference in population proportions pA−pBp_A-p_B. This question assesses understanding of the sampling distribution for the difference in two sample proportions, a key concept in AP Statistics for comparing categorical data from two groups. The sampling distribution of p^A−p^B\hat{p}_A - \hat{p}_B is centered at the true population difference (pA−pB)(p_A - p_B), not at the observed sample difference, and its spread is given by the standard deviation formula that incorporates both population proportions and sample sizes. A common distractor is choice A, which incorrectly suggests the distribution is centered at the observed difference from one sample, confusing the sample statistic with the parameter. In a mini-lesson on difference distributions: when taking independent random samples from two populations, the difference in sample proportions is an unbiased estimator of the true difference, meaning repeated samples will produce values varying around (pA−pB)(p_A - p_B) with a normal shape under large sample conditions. The spread decreases as sample sizes increase, reflecting more precise estimates. This centering at the parameter ensures that inferences about the population difference are valid.
  9. A company compares the proportion of customers who renew a subscription under two email campaigns. From Campaign 1, an SRS of n1=80n_1=80 customers is selected; from Campaign 2, an independent SRS of n2=320n_2=320 customers is selected. The statistic of interest is p^1−p^2\hat{p}_1-\hat{p}_2. Over many repetitions, the sampling distribution of p^1−p^2\hat{p}_1-\hat{p}_2 is centered at p1−p2p_1-p_2 and has standard deviation p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat{p}_1-\hat{p}_2 is centered at p1−p2p_1-p_2. This question evaluates knowledge of the sampling distribution for differences in sample proportions, essential in AP Statistics for comparing success rates between groups like subscription renewals. The distribution is centered at (p1p_1 - p2p_2) with spread determined by (p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}), which accounts for unequal sample sizes without the larger one dominating entirely. Choice E is a distractor, wrongly claiming the center is the observed difference, which varies per sample while the true center remains the population parameter. Mini-lesson: the sampling distribution of (p^1−p^2\hat{p}_1 - \hat{p}_2) from independent samples models how this statistic behaves over many repetitions, always centering on the fixed (p1p_1 - p2p_2) but with variability that combines the uncertainties from each sample. Larger samples reduce spread, improving reliability for hypothesis tests or confidence intervals. Even with (n2n_2 > n1n_1), both contribute to the overall variance.
  10. To compare two products, a store samples customers and records whether each customer makes a purchase. Product A: independent random sample of nA=250n_A=250 customers; Product B: independent random sample of nB=250n_B=250 customers. The statistic is p^A−p^B\hat{p}_A-\hat{p}_B. Which statement is correct about the effect of independence on the sampling distribution?
    Answer: Independence allows the variances from the two samples to add when finding the standard deviation of p^A−p^B\hat{p}_A-\hat{p}_B. This question explores the role of independence in the sampling distribution of proportion differences, essential in AP Statistics for correct variance calculations. Independence allows adding the variances of the individual proportions to get the SD of the difference, ensuring accurate spread quantification. Choice A is a distractor, wrongly stating independence makes the statistic exactly equal the parameter every time, ignoring sampling variability. Mini-lesson: for independent samples, the distribution of (\hat{p}_A - \hat{p}_B) is centered at (pAp_A - pBp_B) with SD from added variances, approximately normal under conditions, but without independence, covariance terms complicate it. This doesn't imply equal proportions or change the shape to uniform. Independence is key for valid two-sample inferences.
  11. A researcher compares the proportion of plants that survive under two fertilizers. Fertilizer A is tested on a random sample of n1=100n_1=100 plants and Fertilizer B on an independent random sample of n2=25n_2=25 plants; p^1\hat p_1 and p^2\hat p_2 are the sample survival proportions. Consider the sampling distribution of p^1−p^2\hat p_1-\hat p_2 over repeated experiments. Which statement is correct?
    Answer: Because n2n_2 is smaller, the sampling distribution of p^1−p^2\hat p_1-\hat p_2 will typically have greater spread than if n2n_2 were larger (all else equal). This question focuses on how sample size affects the spread of the sampling distribution. The standard error formula p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} shows that smaller sample sizes lead to larger standard errors. Since n2=25n_2 = 25 is relatively small, it contributes more to the overall spread than if it were larger. The spread depends on both sample sizes, not just the larger one (eliminating B). The given formula in C uses sample proportions instead of population proportions (eliminating C). Equal population proportions don't eliminate variability (eliminating D). Independence doesn't determine the sign of the difference (eliminating E).
  12. A city compares the proportion of households that recycle weekly in two neighborhoods. Neighborhood 1: SRS of n1=100n_1=100 households; Neighborhood 2: independent SRS of n2=100n_2=100 households. The statistic is p^1−p^2\hat{p}_1-\hat{p}_2. Which statement is correct about the standard deviation of the sampling distribution of p^1−p^2\hat{p}_1-\hat{p}_2?
    Answer: It is p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}. This question focuses on the standard deviation formula for the sampling distribution of proportion differences, a core AP Statistics tool for quantifying variability in comparisons. The correct spread is (p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}), depending on both population proportions and sample sizes. Choice C distracts by using sample proportions for the population distribution, which is for estimation, not the true SD. Mini-lesson: the difference distribution from independent samples centers at (p1p_1 - p2p_2) with this SD formula, which doesn't zero out for equal n or depend solely on the difference. It separates effects of p and n, guiding sample size planning. Understanding this prevents underestimating variability in equal-sized groups.
  13. A nonprofit compares the proportion of donors who give again within a year between two outreach methods. An independent random sample of n1=70n_1=70 donors contacted by email and n2=130n_2=130 donors contacted by phone is selected; p^1\hat p_1 and p^2\hat p_2 are the sample proportions who donate again. Over many repetitions, the sampling distribution of p^1−p^2\hat p_1-\hat p_2 is examined. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat p_1-\hat p_2 has mean p1−p2p_1-p_2, regardless of the sample sizes, assuming random sampling and independence. This question tests the fundamental property about the mean of a sampling distribution. The sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 has mean equal to p1−p2p_1 - p_2 regardless of sample sizes, as long as we have random sampling and independence. This is a key unbiased property of the difference in sample proportions. The mean is not the sample statistic itself (eliminating B). The mean is only 0 when p1=p2p_1 = p_2, not just because both are proportions (eliminating C). Independence doesn't eliminate variability (eliminating D). The normality condition requires checking success-failure conditions, not just having n1+n2≥30n_1 + n_2 \geq 30 (eliminating E).
  14. A researcher compares the proportion of plants that survive under two fertilizers. Fertilizer A: random sample of nA=40n_A=40 plants; Fertilizer B: independent random sample of nB=40n_B=40 plants. The statistic is p^A−p^B\hat{p}_A-\hat{p}_B. Consider the condition for using a normal approximation for the sampling distribution of p^A−p^B\hat{p}_A-\hat{p}_B. Which statement is correct?
    Answer: A normal approximation is appropriate if each sample has at least 10 expected successes and 10 expected failures: nApA, nA(1−pA), nBpB, nB(1−pB)≥10n_Ap_A,\,n_A(1-p_A),\,n_Bp_B,\,n_B(1-p_B)\ge 10. This question assesses conditions for normality in the sampling distribution of proportion differences, vital in AP Statistics for valid inferences. The normal approximation holds if each group has at least 10 expected successes and failures, ensuring the individual proportion distributions are mound-shaped. Choice C is a distractor, oversimplifying to total sample size >=30 without checking success-failure counts per group. Mini-lesson: for (\hat{p}_A - \hat{p}_B) from independent samples, normality requires (n p geq 10) and (n(1-p) geq 10) for each, allowing the difference to be approximately normal centered at (pAp_A - pBp_B) with calculable spread. This isn't guaranteed by equal proportions or observed values alone. Proper checks prevent skewed distributions in analysis.
  15. A school district wants to compare support for a new start-time policy between two groups of parents. A random sample of n1=80n_1=80 elementary-school parents and an independent random sample of n2=120n_2=120 high-school parents are surveyed; in each group, the sample proportion who support the policy is recorded as p^1\hat p_1 and p^2\hat p_2. If these sampling methods were repeated many times, the distribution of p^1−p^2\hat p_1-\hat p_2 would be approximately normal with mean p1−p2p_1-p_2 and standard deviation p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}} (assuming conditions are met). Which statement is correct?
    Answer: The mean of the sampling distribution of p^1−p^2\hat p_1-\hat p_2 is p1−p2p_1-p_2. This question tests understanding of the sampling distribution of the difference in sample proportions. The sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 describes how this difference varies across many repeated samples. Its mean (center) is the true population difference p1−p2p_1 - p_2, not the observed sample difference from one particular sample (eliminating A). The distribution does have variability even with fixed sample sizes because different samples yield different proportions (eliminating B). The standard deviation formula uses population proportions p1p_1 and p2p_2, not sample proportions (eliminating D). The center is p1−p2p_1 - p_2 regardless of whether sample sizes are equal (eliminating E).
  16. A city compares the proportion of residents who support building a new park in two neighborhoods. An SRS of n1=40n_1=40 residents from Neighborhood 1 and an independent SRS of n2=40n_2=40 residents from Neighborhood 2 are taken; p^1\hat p_1 and p^2\hat p_2 are the sample proportions in favor. The sampling distribution of p^1−p^2\hat p_1-\hat p_2 is considered over many repetitions. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat p_1-\hat p_2 is centered at the population difference p1−p2p_1-p_2. This question tests the fundamental property that sampling distributions are centered at population parameters. The sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2 describes the behavior of this statistic over many repetitions and is centered at the true population difference p1−p2p_1 - p_2. It is not centered at any particular sample's observed difference (eliminating B). The standard deviation is positive even when asking the same question, due to sampling variability (eliminating C). The distribution can be described theoretically without knowing specific sample values (eliminating D). Normality requires meeting conditions, not just comparing two proportions (eliminating E).
  17. A school district wants to compare the proportion of students who prefer online homework in two middle schools. An SRS of n1=150n_1=150 students is taken from School 1 and an independent SRS of n2=150n_2=150 students is taken from School 2. The statistic p^1−p^2\hat{p}_1-\hat{p}_2 is computed. If the sampling is repeated many times, the distribution of p^1−p^2\hat{p}_1-\hat{p}_2 will have a mean of p1−p2p_1-p_2 and a standard deviation that decreases as the sample sizes increase. Which statement is correct?
    Answer: If both sample sizes were doubled, the sampling distribution of p^1−p^2\hat{p}_1-\hat{p}_2 would typically be less spread out. This question tests comprehension of how sample size affects the sampling distribution of the difference in proportions, a fundamental AP Statistics skill for designing comparative studies. Doubling both sample sizes reduces the spread of the distribution while keeping the center at (p1p_1 - p2p_2), as the standard deviation formula shows an inverse relationship with (n). Choice B distracts by suggesting increased spread with larger samples, which is incorrect since larger samples provide more precise estimates and less variability. Mini-lesson: the difference (p^1−p^2\hat{p}_1 - \hat{p}_2) from independent samples has a sampling distribution that's approximately normal, centered at the true difference, with spread shrinking as sample sizes grow, enabling tighter confidence intervals. Equal sample sizes don't eliminate variability; there's always sampling error unless populations are fully enumerated. This principle guides researchers in balancing precision with cost.
  18. A researcher compares the proportion of commuters who bike to work in two cities. City 1: SRS of n1=120n_1=120; City 2: independent SRS of n2=180n_2=180. The statistic is p^1−p^2\hat{p}_1-\hat{p}_2. In repeated sampling, the sampling distribution is approximately normal under appropriate conditions and centered at p1−p2p_1-p_2. Which statement is correct?
    Answer: The sampling distribution of p^1−p^2\hat{p}_1-\hat{p}_2 can include both positive and negative values even if p1−p2≠0p_1-p_2\ne 0. This question tests understanding of variability in the sampling distribution of proportion differences, important in AP Statistics for interpreting possible outcomes. Even if (p1p_1 - p2p_2 eq 0), sampling error allows (p^1−p^2\hat{p}_1 - \hat{p}_2) to be positive or negative, reflecting the distribution's spread around the center. Choice A distracts by claiming a negative parameter forces negative statistics every time, underestimating variability. Mini-lesson: the difference distribution from independent samples centers at (p1p_1 - p2p_2) (which can be negative) with normal shape, but values span both sides due to chance, not biased by sample size differences. Proportions' nonnegativity doesn't restrict the difference's sign. This variability underpins confidence intervals crossing zero.
  19. Two independent random samples are taken to compare the proportion of voters who support a ballot measure in two counties. County X: nX=60n_X=60; County Y: nY=60n_Y=60. The statistic is p^X−p^Y\hat{p}_X-\hat{p}_Y. Suppose the true population proportions are pX=0.50p_X=0.50 and pY=0.50p_Y=0.50. Over many repetitions, which statement is correct about the sampling distribution of p^X−p^Y\hat{p}_X-\hat{p}_Y?
    Answer: Its mean is 0, but it will still vary from sample to sample. This question probes the properties of the sampling distribution for proportion differences when population proportions are equal, relevant in AP Statistics for null hypothesis scenarios. With (pXp_X = pYp_Y = 0.5), the center is 0, but spread exists due to sampling variability, so (\hat{p}_X - \hat{p}_Y) fluctuates around 0 in repeated samples. Choice B is a distractor, falsely implying no variability when the mean is 0, ignoring that even equal proportions yield differing sample outcomes by chance. Mini-lesson: for independent samples, the distribution of (p^1−p^2\hat{p}_1 - \hat{p}_2) is centered at (p1p_1 - p2p_2) (here 0) with standard deviation reflecting sample sizes and proportions, always showing spread unless samples are infinite. This variability is key for understanding p-values in tests of equal proportions. Proportions being between 0 and 1 doesn't force the difference to be positive.
  20. A tech company compares the proportion of users who enable two-factor authentication on two platforms. Platform W: SRS of nW=1000n_W=1000 users; Platform M: independent SRS of nM=1000n_M=1000 users. The statistic is p^W−p^M\hat{p}_W-\hat{p}_M. Suppose pWp_W and pMp_M stay the same over time and the sampling method stays the same. Which statement is correct about what would happen to the sampling distribution if both sample sizes were reduced to nW=nM=100n_W=n_M=100?
    Answer: The sampling distribution would have the same center but typically larger standard deviation (more spread). This question investigates how changing sample sizes affects the sampling distribution of proportion differences, a practical AP Statistics concept for study design adjustments. Reducing sizes keeps the center at (pWp_W - pMp_M) but increases spread, as the SD grows with smaller n in the formula. Choice C is a distractor, incorrectly stating smaller samples reduce variability, when the opposite is true due to less information. Mini-lesson: for independent samples, the difference distribution's center is invariant to sample size, but spread inversely relates to n, so halving sizes widens it without shifting to observed values or zero. Populations staying the same doesn't negate size effects. This informs trade-offs in precision versus feasibility.

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