CAIIB ABM Module A Unit 4 MCQs – Correlation and Regression

Correct Answer: B. To measure the strength and direction of the linear association between the variables. This option focuses specifically on what correlation analysis aims to achieve.

Correct Answer: C. A direct relationship between the variables. This option uses clearer terms like “one variable increases as the other variable also increases” instead of relying on “high values.”

Correct Answer: D. An absence of correlation between the variables. This option replaces “no relationship” with “an absence of correlation” for better clarity.

Correct Answer: B. The variables have a strong linear relationship. This option uses “strong linear relationship” which is more descriptive.

Correct Answer: B. To measure the strength and direction of a linear relationship between two variables. This option elaborates on the function of ‘r’ more clearly.

Correct Answer: C. -2.85. Covariance (X, Y) = (∑xy / N) – (mean of X * mean of Y) = (180 / 7) – (5 * 5.71) = 25.71 – 28.55 = -2.84. Closest option is -2.85

Correct Answer: B. 3.78. Standard deviation of X = √((∑x²/N) – (mean of X)²) = √((275/7) – (5)²) = √(39.29 – 25) = √14.29 = 3.78

Correct Answer: B. 1. Correlation coefficient = Covariance / (Standard deviation of X * Standard deviation of Y) = 50 / (5 * 10) = 50 / 50 = 1

Correct Answer: D. -0.95. The strength is determined by the absolute value. |-0.95| is the greatest.

Correct Answer: C. The variables do not have a linear relationship. This is a more precise way of stating the absence of a linear relationship.

Correct Answer: B. The average of the products of deviations of X and Y from their means. Covariance, denoted as cov(X,Y), is a measure of how much two variables change together. It is calculated as the sum of the products of the differences of each variable from its mean, divided by the number of observations. For example, if you have data for the number of hours studied (X) and marks obtained (Y) for a set of students, the covariance would indicate whether students who study more tend to get higher marks.

Correct Answer: C. -1 and 1. The value of the correlation coefficient ‘r’ is always between -1 and 1, inclusive. A value of 1 indicates a perfect positive linear relationship, a value of -1 indicates a perfect negative linear1 relationship, and a value of 0 indicates no linear relationship.2 For instance, if the price of a commodity increases, and the demand decreases proportionally, the correlation coefficient would be -1. If they move in the same direction proportionally, it would be 1.

Correct Answer: C. There is a perfect inverse linear relationship between the variables. A correlation coefficient of -1 means that as one variable increases, the other variable decreases by a proportional amount, and all the data points lie exactly on a straight line with a negative slope. An example could be the relationship between the number of hours spent watching television and the hours spent studying for an exam, assuming a perfect inverse relationship exists where increased TV time directly leads to decreased study time.

Correct Answer: A. 128.23. The covariance cov(X,Y) is calculated as ¹⁄₇∑(x−x̄)(y−ȳ). Given ∑(x−x̄)(y−ȳ) = 897.61 and N=7, the covariance is 897.61⁄7 = 128.23. For example, if X is the number of units produced and Y is the cost, and for 7 observations, the sum of the products of deviations from the mean is 897.61, the average product of deviations, or covariance, is 128.23.

Correct Answer: B. 16.77. The standard deviation of X (σₓ) is calculated as √(∑(x−x̄)²⁄N). Given ∑(x−x̄)² = 1967.9 and N=7, σₓ = √(1967.9⁄7) = √281.13 ≈ 16.77. For example, if X represents the heights of 7 students and the sum of the squared differences from the mean height is 1967.9, the standard deviation of their heights is approximately 16.77 units.

Correct Answer: A. 0.8383. The correlation coefficient ‘r’ is calculated as r = cov(X,Y) ⁄ (σₓσᵧ). Given cov(X,Y)=128.23, σₓ=19.84, and σᵧ=7.71, the correlation coefficient is r = 128.23 ⁄ (19.84 × 7.71) = 128.23 ⁄ 152.97 ≈ 0.8383. This value close to 1 indicates a strong positive linear relationship between the two variables. For instance, if X is rainfall and Y is crop yield, a coefficient of 0.8383 suggests that higher rainfall is strongly associated with higher crop yields.

Correct Answer: B. To find the equation of the relationship between the variables. Regression analysis helps to know the relationship between two variables, and to find the equation of this relationship, a line that best represents the points is drawn.

Correct Answer: B. To minimise the difference between observed and estimated values. The technique of least squares is used to ensure that the difference between the points in the scatter diagram and those on the line is minimal.

Correct Answer: C. The slope of the line. ‘b’ in the equation y=a+bx as the slope of the line.

Correct Answer: D. The y-intercept. ‘a’ as the y-intercept, which is the point where the line crosses the y-axis.

Correct Answer: B. The prediction is reasonably accurate. A small value of Se indicates that our estimates are fairly accurate.

Correct Answer: B. 62.77. Using the given equation y=39.96+0.3258x, when x=70, y=39.96+0.3258×70=39.96+22.81=62.77.

Correct Answer: A. 0.3258. The formula for ‘b’ is b=cov(X,Y)⁄σₓ². Given cov(X,Y)=128.23 and σₓ²=393.58, b=128.23⁄393.58=0.3258.

Correct Answer: C. 39.96. The formula for ‘a’ is a=ȳ−bx̄. Given ȳ=61, b=0.3258, and x̄=64.57, a=61−(0.3258×64.57)=61−21.04=39.96.

Correct Answer: A. (57.77, 67.77). 65 per cent confidence interval for the actual value of y lies between (ŷ−Se) and (ŷ+Se). Given ŷ = 62.77 and Se=5, the interval is (62.77−5, 62.77+5)=(57.77, 67.77).

Correct Answer: B. (52.77, 72.77). 95 per cent confidence interval for the actual value of y lies between (ŷ−2Se) and (ŷ+2Se). Given ŷ = 62.77 and Se=5, the interval is (62.77−2×5, 62.77+2×5)=(62.77−10, 62.77+10)=(52.77, 72.77).

Correct Answer: B. Linear relationships. The coefficient of correlation is specifically designed to quantify the strength and direction of a straight-line association between two variables, not curved or other types of relationships. For example, it assesses if points on a scatter plot tend to fall along a straight line.

Correct Answer: C. 0. Even if variables have a clear pattern, if that pattern is not linear (like a circle), the coefficient of correlation, which measures linear association, will be zero. This signifies the absence of a linear trend, not necessarily the absence of any relationship.

Correct Answer: C. Only linear relationships. Correlation analysis should be applied only to linear relationships because the coefficient measures the degree of linear association. Applying it to non-linear patterns can be misleading.

Correct Answer: C. Homogeneous. The data used for correlation analysis should be homogeneous, meaning it should come from a single, consistent source or population to ensure the calculated correlation reflects a genuine relationship within that group.

Correct Answer: A. Spurious correlation. This occurs when a correlation seems to exist in a combined dataset drawn from different groups (heterogeneous), but this correlation does not exist within the individual groups themselves. For example, combining shoe size data from children and adults might show a spurious correlation with reading ability.

Correct Answer: C. Strong direct. A value near +1 signifies that the two variables have a strong tendency to increase together in a linear fashion. For example, height and weight in adults often show a strong direct correlation.

Correct Answer: C. No, it only indicates association. A high correlation shows that variables move together, but it does not explain why. For instance, ice cream sales and crime rates might be positively correlated, but both are likely caused by a third variable (hot weather), not one causing the other.

Correct Answer: B. Correlation does not imply causation. Although the two variables show a strong positive correlation, it is illogical to conclude that one causes the other; other factors likely contribute to the increase in both phenomena independently.

Correct Answer: C. To get an initial visual sense of the relationship between two variables. A scatter diagram plots observed data points and helps visually assess the pattern, direction, and strength of the potential relationship between variables before performing calculations.

Correct Answer: D. Independent variable. The independent variable is the one whose values are known or controlled and used to predict the values of the other variable. For example, using years of experience (independent) to predict salary (dependent).

Correct Answer: C. Dependent variable. The dependent variable is the outcome or result that the analysis aims to predict based on the value(s) of the independent variable(s). For example, predicting crop yield (dependent) based on rainfall (independent).

Correct Answer: C. Regression analysis. Regression analysis specifically deals with finding the mathematical equation (often a line) that best describes the relationship between a dependent variable and one or more independent variables.

Correct Answer: C. How accurate the predictions from a regression equation are likely to be. It quantifies the typical deviation between the observed values of the dependent variable and the values predicted by the regression model. A smaller standard error indicates better predictive accuracy.

Correct Answer: B. The strength of the relationship. Correlation analysis focuses on quantifying the degree and direction (positive or negative) of the linear association between two variables.

Correct Answer: C. Coefficient of correlation. The coefficient of correlation (often denoted as ‘r’) is the specific statistical measure used to indicate the strength and direction of a linear relationship between two variables.

Correct Answer: D. It cannot be calculated if the sample size is small.

Correct Answer: B. There is no linear relationship between the variables. A correlation coefficient of zero specifically indicates the absence of a linear association, but a non-linear relationship (like curvilinear) might still exist.

Correct Answer: C. Trend Analysis. Trend analysis involves examining patterns in data over time or across variables to identify the nature of the relationship, such as whether it is linear, curvilinear, etc..

Correct Answer: C. Spurious correlation. Combining heterogeneous data can create an artificial correlation that doesn’t truly represent the relationship within the individual, homogeneous subgroups.

Correct Answer: C. -1 to +1. The coefficient of correlation ‘r’ is always bounded between -1 (perfect inverse linear relationship) and +1 (perfect direct linear relationship), inclusive.

Correct Answer: D. Confusing correlation with causation. A strong correlation merely shows association. Attributing causality without further evidence, especially when a lurking variable might be involved, is a common statistical fallacy.

Correct Answer: C. Coefficient of correlation. The definition of the coefficient of correlation involves covariance in the numerator, measuring how two variables change together, standardized by their respective standard deviations.

Correct Answer: C. To determine if a statistical relationship exists between the number of weekend events and the number of weekend accidents. The goal was to see if changes in one variable (events) corresponded systematically with changes in the other variable (accidents), specifically within the weekend timeframe.

Correct Answer: C. To improve the accuracy of product pricing and budget for costs at different production levels. Accurate cost estimation is crucial for setting prices correctly and for financial planning, especially when costs don’t follow a simple fixed or variable pattern.

Correct Answer: C. It generally provides more reliable predictions by mathematically modeling the relationship and allows for quantifying the expected prediction error. Statistical models aim to capture the underlying trend, leading to better estimates than guesses, and also provide measures (like standard error) of how much the actual values might deviate from the prediction.

Correct Answer: B. Regression analysis. Regression analysis is the statistical method used specifically to model the relationship between a dependent variable (like overhead costs) and one or more independent variables (like production units) by fitting an equation (often a line) to the observed data.

Correct Answer: B. The number of accidents recorded when 10 events occurred. In the pair (X, Y) representing (Events, Accidents), the second value corresponds to the measurement of the Y variable (Accidents) when the X variable (Events) was at the level specified by the first value.

Correct Answer: C. 191. The question provides only one specific data pairing (40 units, cost 191) and asks what cost corresponds to that specific instance according to the provided statement.

Correct Answer: C. Factors other than just the production unit count likely influence overhead costs, or there is inherent variability. If the relationship was perfectly simple and deterministic with production units as the sole driver, the cost should be identical for the same number of units. Variation implies other factors or randomness are involved.

Correct Answer: C. To identify the peak production level achieved during the observed period. The maximum value in a dataset represents the highest point reached for that variable within the scope of the observations.

Correct Answer: C. The minimum overhead cost recorded during that specific observation period. The minimum value in a dataset indicates the lowest point observed for that variable within the set of measurements taken.

Correct Answer: C. To predict the expected overhead cost associated with producing 50 units. The regression equation is specifically used for prediction – plugging in a value for the independent variable (production units = 50) yields an estimate for the dependent variable (overhead cost).