Business Analytics Quiz: Correlation And Association
10 questions · exam conditions
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Correlation And AssociationQuestion 1 of 10

An analyst finds a Pearson correlation of −0.62-0.62 between discount percentage and gross profit measured in dollars. The analyst replaces discount percentage with percentage of list price retained, calculated as 100−discount percentage100-\text{discount percentage}, and expresses gross profit in thousands of dollars.

What is the correlation between the two transformed variables?

−0.62-0.62, because changing measurement units cannot affect either the sign or magnitude of a correlation.
0.620.62, because reversing the discount scale changes the sign while rescaling profit leaves correlation unchanged.
−0.0062-0.0062, because converting gross profit to thousands proportionally reduces the reported correlation coefficient.
0.00620.0062, because both reversing the discount scale and rescaling profit alter the magnitude of correlation.
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Business Analytics Quiz

Business Analytics Quiz: Correlation And Association

Practice Correlation And Association in Business Analytics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Correlation And Association, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Analytics.

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.

All questions

Question 1

An analyst finds a Pearson correlation of −0.62-0.62 between discount percentage and gross profit measured in dollars. The analyst replaces discount percentage with percentage of list price retained, calculated as 100−discount percentage100-\text{discount percentage}, and expresses gross profit in thousands of dollars.

What is the correlation between the two transformed variables?

  1. −0.62-0.62, because changing measurement units cannot affect either the sign or magnitude of a correlation.
  2. 0.620.62, because reversing the discount scale changes the sign while rescaling profit leaves correlation unchanged. (correct answer)
  3. −0.0062-0.0062, because converting gross profit to thousands proportionally reduces the reported correlation coefficient.
  4. 0.00620.0062, because both reversing the discount scale and rescaling profit alter the magnitude of correlation.
Explanation: Whenever you see a question about correlation and variable transformations, your first instinct should be to recall how Pearson correlation responds to linear changes. The key rule: multiplying or adding a constant to a variable changes the sign of the correlation only if the multiplier is negative; it never affects the magnitude. Here, two transformations occur. First, discount percentage is replaced by (100−discount%)(100 - \text{discount\%}). This is equivalent to multiplying the original variable by −1-1 (plus a constant, which is irrelevant). That negative multiplier flips the sign of the correlation from −0.62-0.62 to +0.62+0.62. Second, gross profit is divided by 1,000 to convert dollars to thousands. Dividing by a positive constant leaves the sign unchanged and, crucially, does not affect the magnitude of the correlation at all. The result is +0.62+0.62, confirming B is correct. A is wrong because it claims sign cannot change — it can, whenever a variable is multiplied by a negative constant. The sign does flip here. C and D both claim that rescaling profit to thousands shrinks the correlation coefficient numerically (e.g., dividing by 1,000 to get 0.00620.0062). This is a classic trap: correlation is unitless and dimensionless — it is already a standardized measure, so multiplying or dividing either variable by any positive constant has zero effect on its value. D compounds that error by also flipping the sign incorrectly. Study tip: Remember that Pearson correlation is invariant to any positive linear rescaling. Only a negative multiplier changes the sign. Units never change the magnitude.

Question 2

For four comparable stores, local advertising expenditures of 11, 22, 33, and 44 units correspond to sales of 22, 44, 66, and 88 units. A fifth store spent 1010 units on advertising but generated only 11 unit of sales.

How does the fifth store affect the Pearson correlation between advertising and sales?

  1. It produces a negative correlation because the fifth store is a high-leverage point whose outcome contradicts the original trend. (correct answer)
  2. It weakens but does not reverse the correlation because a single observation cannot change the sign of an existing relationship.
  3. It leaves the correlation equal to 11 because the first four stores follow an exact linear relationship.
  4. It drives the correlation to approximately zero because an extreme observation on the horizontal axis always eliminates linear association.
Explanation: When a question involves how a new data point changes correlation, your first instinct should be to think about leverage — how far that point sits from the center of the existing data, and whether its outcome aligns with or contradicts the established trend. Here, the first four stores form a perfect line: advertising (1,2,3,4)(1,2,3,4) maps exactly to sales (2,4,6,8)(2,4,6,8), giving a correlation of r=1r = 1. The fifth store spends 1010 units on advertising — far above the others — but produces only 11 unit of sales, the lowest in the dataset. This is a classic high-leverage outlier: it sits at an extreme position on the horizontal axis and pulls hard against the existing upward trend. When you compute the new correlation, that single contradictory point is powerful enough to drag rr into negative territory, making A the correct answer. Choice B is tempting but wrong — it assumes a single point can only weaken, never reverse, a correlation. In reality, a sufficiently extreme observation can absolutely flip the sign, especially when it contradicts the trend as dramatically as this one does. Choice C incorrectly assumes the perfect relationship among the first four stores is immune to new data; correlation is recalculated across all observations every time you add one. Choice D contains a grain of truth — extreme horizontal values do distort correlations — but "always eliminates" (drives to zero) is too strong and inaccurate; the actual direction depends on where the outlier's outcome falls, not just its position. As a study tip: watch for outliers that combine extreme xx-values with outcome values that contradict the trend — these are the scenarios most likely to reverse, not merely weaken, a correlation.

Question 3

Across all customers, annual household income and spending on premium products have a correlation of 0.580.58. An analyst then examines only customers whose annual incomes fall between 95,00095{,}000 and 100,000100{,}000 and obtains a correlation of 0.120.12.

Which explanation is most defensible based only on this information?

  1. The full-sample correlation must be spurious because a valid relationship would have the same coefficient in every income subset.
  2. Restricting the range of income can reduce observed correlation even if the broader population has a meaningful association. (correct answer)
  3. Premium spending becomes independent of income above 95,00095{,}000 because the subgroup correlation is close to zero.
  4. The subgroup result proves that premium spending causes income differences only outside the selected income interval.
Explanation: Whenever you see a question comparing correlations across different sample sizes or ranges, think about range restriction — a core concept in statistics that explains why a relationship can look weaker in a narrow slice of data than it does in the full population. Here's the key idea: correlation measures how consistently two variables move together across the variation present in your sample. When you restrict your sample to a narrow band of income ($95,000\$95{,}000 to $100,000\$100{,}000), you dramatically reduce the spread of the independent variable. With almost no variation in income to work with, the statistical algorithm has little signal to detect, and the correlation naturally compresses toward zero — even if the true underlying relationship is strong. This is exactly what happened here: the full-sample correlation of 0.580.58 collapses to 0.120.12 simply because the income range is too tight to reveal it. Answer B correctly identifies this mechanism. Answer A is wrong because a valid relationship does not need to produce identical correlations in every subgroup — that's not how statistics works. Subgroup correlations legitimately vary, especially under range restriction. Answer C makes an unjustified causal claim: a near-zero correlation in one narrow income band does not prove that spending and income are independent above $95,000\$95{,}000 — the restricted range is a far simpler explanation. Answer D invents a causal reversal (spending causes income) that the data never support; correlation, in any direction, cannot establish causation. Your study tip: whenever a correlation shrinks dramatically in a subgroup, always ask "Is the range of the key variable restricted here?" before drawing any conclusions about independence or causality.

Question 4

A business analyst reports a Pearson correlation of −0.70-0.70 between average delivery time and customer satisfaction score. The analyst plans to summarize the result using the coefficient of determination.

Which summary is most appropriate for this bivariate relationship?

  1. Approximately 70%70\% of customer dissatisfaction is caused by longer delivery times in the observed data.
  2. Approximately 49%49\% of delivery-time observations are below average when satisfaction scores are above average.
  3. Approximately 30%30\% of satisfaction scores cannot be predicted because the correlation is negative rather than positive.
  4. Approximately 49%49\% of the variation in satisfaction is associated with a linear relationship with delivery time. (correct answer)
Explanation: When you see a Pearson correlation coefficient (r), a key follow-up tool is the coefficient of determination (r2r^2), which tells you how much of the variation in one variable is linearly explained by the other. Here, r=−0.70r = -0.70, so r2=(−0.70)2=0.49r^2 = (-0.70)^2 = 0.49, meaning 49% of the variability in satisfaction scores is associated with a linear relationship with delivery time. That makes D the correct and precise interpretation. Choice A is tempting but commits two serious errors: it replaces "variation explained" with "dissatisfaction caused," introducing causation language that correlation never supports, and it uses 70% (the raw r) instead of 49% (the proper r2r^2). Correlation describes association, not causation. Choice B misreads r2r^2 as a frequency count — the idea that 49% of observations fall below average under certain conditions. The coefficient of determination says nothing about how individual data points are distributed relative to their means; that's a different statistical concept entirely. Choice C invents a rule that doesn't exist: a negative correlation does not reduce the coefficient of determination or leave a portion "unpredictable." Squaring r always produces a non-negative value regardless of direction, so the sign of the correlation is irrelevant to r2r^2. Study tip: Always square r to get r2r^2, and frame your interpretation as "__% of the variation in Y is associated with (or explained by) a linear relationship with X." Watch for distractors that sneak in causal language ("caused by") or misuse the raw correlation value instead of r2r^2.

Question 5

A company calculates the correlation between service calls and customer spending separately for two equally sized regions. The correlation is 0.600.60 in each region. Customers in one region also have much higher average spending and fewer average service calls than customers in the other region.

What can the analyst conclude about the correlation after pooling the two regions?

  1. It must equal 0.600.60 because the regions have equal sample sizes and identical within-region correlations.
  2. It must be greater than 0.600.60 because pooling equal samples always increases the amount of observed variation.
  3. It must remain positive because two positive within-region associations cannot produce a negative pooled association.
  4. It cannot be determined from the two correlations because differences between regional means also affect the pooled correlation. (correct answer)
Explanation: Whenever you see a question about combining groups with different means, think about Simpson's Paradox and the broader phenomenon of ecological aggregation — the idea that pooling subgroups can dramatically shift a statistic beyond what any individual group shows. Here's why D is correct: When two regions have different average levels of both variables (spending and service calls), pooling them introduces between-group variation that wasn't present in either within-group analysis. Imagine plotting both regions together — the cluster of high-spending, low-call customers from one region sits far from the cluster of low-spending, high-call customers from the other. That separation between clusters creates its own trend line in the combined data, which can pull the pooled correlation higher, lower, or even in the opposite direction compared to either region alone. Because you don't know the exact means and standard deviations of each region, you simply cannot calculate the pooled correlation from the two regional correlations alone. A is wrong because equal sample sizes don't neutralize the effect of differing means. Equal n only means each region contributes the same number of data points — it says nothing about where those points sit in the variable space. B is wrong because more variation doesn't guarantee a predictable directional change in correlation, which is already a standardized, scale-free measure. C is the trickiest distractor: two positive within-group correlations can actually produce a negative pooled correlation if the group means are arranged the right way — that's precisely Simpson's Paradox. Study tip: Any time a question mentions subgroups with different means being combined, immediately ask yourself whether between-group differences could be distorting the pooled statistic.

Question 6

A marketing dashboard reports that 12%12\% of customers who received a personalized offer purchased a product, compared with 8%8\% of customers who did not receive the offer. Offers were assigned by sales representatives rather than randomly.

Which statement accurately summarizes both the observed association and its limitation?

  1. Offer recipients had a 44-percentage-point higher purchase rate, but representative targeting may confound the estimated effect of the offer. (correct answer)
  2. Offer recipients had a 5050-percentage-point higher purchase rate, and nonrandom assignment affects only the precision of that estimate.
  3. Offer recipients had a 4%4\% relative increase in purchases, and representative assignment is sufficient to establish a causal treatment effect.
  4. Offer recipients had a 1212-percentage-point lift, and the groups cannot be compared in any meaningful way because assignment was nonrandom.
Explanation: Whenever you see a question combining a statistical comparison with a study design flaw, you need to evaluate two things separately: the math of the observed difference, and what that difference actually means given how the data were collected. Start with the numbers. The purchase rate among offer recipients was 12%12\% versus 8%8\% for non-recipients. The absolute difference is 12%−8%=412\% - 8\% = 4 percentage points. That's the correct way to express the observed gap — straightforward subtraction. Now for the design issue: because sales representatives chose who received offers (not a random process), the two groups likely differ in ways beyond just receiving the offer. Representatives probably targeted customers who already seemed more likely to buy — a classic confound. This means the 44-percentage-point gap may reflect pre-existing differences, not the true effect of the offer itself. Answer A captures both points precisely, making it correct. Answer B makes two errors: 5050 percentage points is wildly wrong (that would require one group at 58%58\% and the other at 8%8\%, or similar), and nonrandom assignment doesn't merely affect precision — it introduces bias, a fundamentally different problem. Answer C misquotes the lift as 4%4\% relative (the relative increase is actually 48=50%\frac{4}{8} = 50\%), and fatally claims nonrandom assignment is sufficient for causal inference — the opposite of the truth. Answer D correctly states the 12%12\% figure, but that's the raw rate for one group, not the "lift." More importantly, nonrandom assignment doesn't make comparisons meaningless — it just limits causal interpretation. Study tip: Always distinguish absolute difference, relative difference, and raw rates — exam questions frequently mix them up to test whether you're paying attention to units and framing.

Question 7

A subscription company studies the relationship between the number of promotional emails sent per month and customer response. Response is highest at a moderate email frequency but lower when customers receive either very few or very many emails. The Pearson correlation between email frequency and response is close to 00.

Which interpretation is most appropriate?

  1. Email frequency and response are unrelated because a correlation near 00 rules out any systematic association.
  2. Email frequency has little causal effect because a near-zero correlation implies that changing frequency will not change response.
  3. A meaningful nonlinear association may exist even though Pearson correlation detects little overall linear association. (correct answer)
  4. The association must be strongly negative because response eventually declines as email frequency becomes high.
Explanation: Whenever you see a question involving Pearson correlation, remember that it measures only linear association — the degree to which two variables move together along a straight line. It is completely blind to curved or U-shaped relationships, which is exactly the trap this question sets. The passage describes a classic inverted-U (curvilinear) pattern: response rises with moderate email frequency and falls at both extremes. When a relationship bends like this, the positive slope on the left and the negative slope on the right cancel each other out mathematically, producing a Pearson r≈0r \approx 0 — even though the association is actually quite strong and systematic. Choice C correctly identifies this: a near-zero Pearson correlation does not mean no association exists; it means no linear association was detected. A nonlinear relationship can be very meaningful and still invisible to Pearson's formula. Choice A is wrong because it treats r≈0r \approx 0 as evidence that the variables are completely unrelated. That conclusion is only valid if you've already confirmed the relationship is linear — which the passage clearly contradicts. Choice B goes further by making a causal claim, but correlation (or its absence) never implies causation, and more importantly, the near-zero rr here reflects the shape of the relationship, not the absence of one. Choice D fabricates a conclusion the data don't support — a strong negative correlation would require response to decline consistently across all frequency levels, not just at high values. Your study tip: always ask whether a relationship could be nonlinear before interpreting a low correlation as "no relationship." Scatter plots are your best diagnostic tool here.

Question 8

A retailer compares purchase rates for campaigns X and Y in two customer segments. In the high-propensity segment, campaign X produces purchases from 99 of 1010 customers, while campaign Y produces purchases from 8080 of 100100 customers. In the low-propensity segment, campaign X produces purchases from 2020 of 100100 customers, while campaign Y produces purchases from 11 of 1010 customers.

Which conclusion is best supported by these results?

  1. Campaign X has the higher purchase rate within each segment, but campaign Y has the higher aggregate rate because campaign assignment differs by segment. (correct answer)
  2. Campaign Y has the higher purchase rate within each segment and in aggregate, so segment membership does not affect the comparison.
  3. Campaign X has the higher aggregate purchase rate, although campaign Y performs better after the customers are separated by segment.
  4. Campaign performance is identical after segment adjustment because both campaigns were evaluated on the same total number of customers.
Explanation: Whenever you see a question presenting group-level data broken into subgroups, you should immediately think about Simpson's Paradox — the phenomenon where a trend that appears in combined data reverses or disappears when the data is separated by subgroup. Let's calculate the rates. In the high-propensity segment, Campaign X converts 9/10=90%9/10 = 90\% versus Campaign Y's 80/100=80%80/100 = 80\% — X wins. In the low-propensity segment, Campaign X converts 20/100=20%20/100 = 20\% versus Campaign Y's 1/10=10%1/10 = 10\% — X wins again. So Campaign X outperforms Y within every segment. Yet in aggregate, Campaign X totals 9+20=299 + 20 = 29 purchases out of 110110 customers (≈26%\approx 26\%), while Campaign Y totals 80+1=8180 + 1 = 81 purchases out of 110110 customers (≈74%\approx 74\%). Campaign Y appears dominant overall. Why? Because Campaign Y was disproportionately assigned to the high-propensity segment, which inflates its aggregate rate. Answer A correctly captures this: X wins each segment, but Y wins in aggregate due to unequal segment assignment. Answer B is wrong because Campaign Y does not have the higher rate within each segment — X does in both. Answer C reverses the truth: X does not have the higher aggregate rate; Y does. Answer D is wrong on two counts — performance is not identical after adjustment, and the distractor fabricates a "same total customers" justification that doesn't resolve the paradox. When you see subgroup data, always calculate rates at both levels. If a campaign looks better overall but worse in every subgroup, you're looking at Simpson's Paradox — and the segmented analysis is the more honest comparison.

Question 9

A company observes that employees who voluntarily enroll in an advanced analytics course subsequently receive higher average performance ratings than employees who do not enroll. Course enrollment and later performance rating have a positive correlation.

Which statement best describes what the company can infer from this association?

  1. The course improved performance because performance ratings were measured after enrollment rather than before enrollment.
  2. The course may improve performance, but prior motivation or ability could influence both enrollment and later ratings. (correct answer)
  3. The course cannot improve performance because voluntary enrollment prevents any useful comparison between the employee groups.
  4. The correlation is invalid unless enrollment and performance ratings are measured using the same numerical scale.
Explanation: Whenever you see a question linking an observed association to a causal conclusion, your first instinct should be to ask: could something else explain this pattern? This is the core of confounding variable reasoning, one of the most tested concepts in business analytics. Here, employees who choose to enroll in the analytics course likely differ from non-enrollees before the course even begins — they may be more motivated, more ambitious, or already more skilled. These pre-existing traits could independently drive both the decision to enroll and higher performance ratings later. That's exactly what B captures: the course may help, but you can't isolate its effect because self-selection creates a confound. Voluntary enrollment means the two groups aren't comparable from the start, so the correlation alone doesn't establish causation. A is wrong because the timing of measurement — after enrollment rather than before — doesn't eliminate confounding. The post-enrollment design would only establish causation in a controlled experiment, not an observational study like this one. C goes too far in the opposite direction; voluntary enrollment doesn't make the data useless, it just limits causal inference. You can still generate hypotheses and observe patterns. D is a distractor built on a false measurement rule — there's no requirement that two variables share the same numerical scale to produce a valid correlation. A reliable strategy: when you see correlation data from a self-selected group, always ask "who opted in, and why?" If enrollment wasn't random, a confounding variable is almost certainly lurking.

Question 10

Over 3636 consecutive months, a retailer's advertising expenditure and sales revenue have a correlation of 0.910.91. During the same period, both series generally increased because the retailer opened new locations and prices rose.

Which additional analysis would best assess whether the reported association mainly reflects the shared upward trend?

  1. Recalculate the correlation after expressing both series in dollars rather than converting either series to another unit.
  2. Square the correlation so that the direction of the shared trend no longer affects the strength measure.
  3. Compare month-to-month changes or remove the time trends before recalculating the association between the series. (correct answer)
  4. Exclude all months in which both advertising and revenue increased because those observations create positive correlation.
Explanation: Whenever you see a high correlation between two variables measured over time, your first instinct should be to ask: are these truly related, or do they just both happen to trend upward together? This is the classic spurious correlation problem driven by a shared time trend. When two series both climb steadily over months or years — here, due to new store openings and rising prices — their correlation will naturally be high even if advertising has no real effect on sales. The r=0.91r = 0.91 may be capturing nothing more than "both numbers got bigger over time." The remedy is to isolate the relationship after removing that shared trend. You can do this by analyzing month-to-month changes (first differences) or by detrending both series and correlating the residuals. If the association weakens substantially, the original correlation was mostly trend-driven. This is exactly what C proposes, making it the correct answer. A is wrong because changing units (dollars vs. another denomination) has no effect on correlation — it is scale-invariant by construction. You'd get the same rr regardless. B is wrong because squaring the correlation to get r2r^2 only removes information about direction; it does nothing to address whether a shared trend is inflating the magnitude of the association. D is wrong and statistically destructive — deliberately removing observations where both variables increased would bias your sample and tell you nothing meaningful about the trend problem. Study tip: On any question pairing two time-series with a suspiciously high correlation, immediately flag the possibility of a spurious trend relationship and look for answers that involve differencing, detrending, or controlling for time.