PRC-2 · Chapter 10
Correlation and Regression MCQs with Answers
60 multiple-choice questions on Correlation and Regression for PRC-2 Quantitative Analysis for Business. Try each one before revealing the answer and explanation.
Practise this chapter interactivelyQuestion 1
What is the primary function of a scatter diagram in statistical analysis?
- A) To prove absolute causation between two variables.
- B) To show the visual relation between variables, which might indicate a relationship where there is none.
- C) To calculate the exact variance of a population.
- D) To display the sampling distribution of the mean.
Show answer & explanation
Answer: B) To show the visual relation between variables, which might indicate a relationship where there is none.
A scatter diagram shows the visual relation between variables, but it is important to note that it might indicate a relationship where there is fundamentally none.
Question 2
What is the mathematically defined range for the value of the correlation coefficient (r)?
- A) Exactly 0 to 1
- B) -1 to 0
- C) -1 to +1
- D) 1 to 100
Show answer & explanation
Answer: C) -1 to +1
The value of the correlation coefficient is bounded and always lies strictly in the range -1 to +1.
Question 3
If the correlation coefficient (r) between two variables is 0.9, and a constant value of 3 is subtracted from each observation of variable Y, what will happen to the correlation coefficient?
- A) It will decrease by exactly 3 units.
- B) It will decrease by less than 3 units.
- C) It will remain unchanged.
- D) It will become negative.
Show answer & explanation
Answer: C) It will remain unchanged.
The correlation coefficient remains completely unchanged if a constant value is subtracted from each observation of a variable.
Question 4
If the correlation coefficient (r) is 0.9, and each observation of variable X is multiplied by a constant factor of 100, what is the new value of 'r'?
- A) 90
- B) 0.009
- C) 0.9
- D) Cannot be determined
Show answer & explanation
Answer: C) 0.9
The correlation coefficient remains completely unchanged even if each observation of a variable is multiplied by a large constant.
Question 5
Under which mathematical condition is the correlation between two variables considered to be at its strongest?
- A) When r = 0
- B) When r = 0.5
- C) When r = 1
- D) When r = -0.5
Show answer & explanation
Answer: C) When r = 1
The correlation between two variables is mathematically considered more strong when r = 1, indicating a perfect positive linear relationship.
Question 6
If a researcher adds 3 to each observation of X, and subtracts 5 from each observation of Y, what happens to the original correlation coefficient of 0.85?
- A) It decreases by 2 units.
- B) It increases by 2 units.
- C) It remains exactly 0.85.
- D) It becomes exactly 0.
Show answer & explanation
Answer: C) It remains exactly 0.85.
The correlation coefficient (r) is independent of the change of origin. Adding or subtracting constants to either variable X or Y leaves 'r' completely unchanged.
Question 7
Which of the following scenarios best describes a negative correlation coefficient?
- A) As variable X increases, variable Y tends to increase.
- B) As variable X increases, variable Y tends to decrease.
- C) Variables X and Y do not change.
- D) Variable Y remains constant regardless of X.
Show answer & explanation
Answer: B) As variable X increases, variable Y tends to decrease.
A negative correlation indicates an inverse relationship between variables. As one variable moves in a positive direction, the other tends to move in a negative direction.
Question 8
Given the mathematical data set: ∑x = 75, ∑y = 225, ∑xy = 2851, ∑x² = 625, and n = 6. If calculated, what does the resulting equation y = a + bx represent?
- A) The correlation coefficient.
- B) The standard error of the mean.
- C) The regression line of y on x.
- D) The probability distribution.
Show answer & explanation
Answer: C) The regression line of y on x.
These standard sum parameters are used to calculate the slope (b) and y-intercept (a) to formulate the regression line y on x, which predicts the dependent variable y.
Question 9
What does a correlation coefficient of exactly 0 imply about a scatter diagram?
- A) All points form a perfect upward straight line.
- B) All points form a perfect downward straight line.
- C) The points are completely scattered with no linear trend.
- D) The points form a perfect circle.
Show answer & explanation
Answer: C) The points are completely scattered with no linear trend.
An 'r' value of 0 means there is absolutely no linear relationship between the variables, resulting in a scatter diagram with points dispersed randomly without forming a trendline.
Question 10
In a simple linear regression equation represented as y = 39 - 0.12x, what does the value -0.12 represent?
- A) The y-intercept
- B) The correlation coefficient
- C) The slope or rate of change
- D) The standard deviation
Show answer & explanation
Answer: C) The slope or rate of change
In the linear regression format y = a + bx, the 'b' value (-0.12) is the slope, indicating how much 'y' is expected to decrease for every one-unit increase in 'x'.
Question 11
Why is it important to remember that a high correlation coefficient does not prove causation?
- A) Because correlation coefficients are frequently calculated incorrectly.
- B) Because scatter diagrams might visually indicate a relationship where there fundamentally is none in reality.
- C) Because the standard error cancels out causation.
- D) Because regression lines only work for negative numbers.
Show answer & explanation
Answer: B) Because scatter diagrams might visually indicate a relationship where there fundamentally is none in reality.
Statistical correlation simply measures how variables move together. A scatter diagram might mathematically indicate a strong relationship where there is no actual logical causation between the events.
Question 12
If variable X represents advertising spend and variable Y represents sales revenue, what is the primary purpose of constructing a regression line of Y on X?
- A) To establish the median advertising spend.
- B) To predict expected sales revenue based on a given advertising budget.
- C) To calculate the variance of sales.
- D) To ensure the correlation remains at +1.
Show answer & explanation
Answer: B) To predict expected sales revenue based on a given advertising budget.
The entire purpose of the regression line equation y on x is forecasting. It allows a business to predict the value of the dependent variable (sales) based on the independent variable (advertising spend).
Question 13
If the Pearson correlation coefficient between two variables is calculated as +1.25, what conclusion should be drawn?
- A) There is a perfectly strong positive relationship.
- B) The variables are completely independent.
- C) A calculation error has occurred.
- D) The standard error is unusually high.
Show answer & explanation
Answer: C) A calculation error has occurred.
Because the absolute mathematical range for any valid correlation coefficient lies strictly between -1 and +1, a result of +1.25 is impossible and indicates a mathematical calculation error.
Question 14
Which of the following statistics specifically measures the proportion of the total variance in the dependent variable that is explained by the independent variable?
- A) The coefficient of determination (r-squared)
- B) The correlation coefficient (r)
- C) The standard error
- D) The level of significance
Show answer & explanation
Answer: A) The coefficient of determination (r-squared)
The coefficient of determination, found by squaring 'r', gives the exact percentage of variation in 'y' that is mathematically explained by the linear relationship with 'x'.
Question 15
When constructing a simple scatter diagram, on which axis is the independent variable (x) traditionally plotted?
- A) The vertical y-axis
- B) The horizontal x-axis
- C) The secondary y-axis
- D) The depth z-axis
Show answer & explanation
Answer: B) The horizontal x-axis
In standard cartesian coordinate mapping for regression and scatter diagrams, the independent variable (the cause) is always plotted along the horizontal x-axis.
Question 16
In basic probability theory, if two events A and B are considered 'mutually exclusive', what is the mathematical probability of them both occurring at exactly the same time?
- A) Exactly 0.5
- B) Exactly 1.0
- C) Exactly 0
- D) The sum of P(A) and P(B)
Show answer & explanation
Answer: C) Exactly 0
Events are mutually exclusive if they cannot happen simultaneously. Therefore, the probability of both occurring together, P(A ∩ B), is zero.
Question 17
A fair six-sided die is rolled. What is the precise probability of rolling a number that is strictly greater than 4?
- A) 1/6
- B) 1/3
- C) 1/2
- D) 2/3
Show answer & explanation
Answer: B) 1/3
On a six-sided die, the numbers strictly greater than 4 are 5 and 6. This represents 2 successful outcomes out of 6 possible total outcomes (2/6), which simplifies to 1/3.
Question 18
If the probability of a specific event occurring (P) is 0.25, what is the mathematical probability of that event NOT occurring?
- A) 0
- B) 0.50
- C) 0.75
- D) 1.00
Show answer & explanation
Answer: C) 0.75
The sum of the probabilities of all possible outcomes in a sample space must equal 1. Therefore, the probability of an event NOT occurring is 1 - P(Event), which is 1 - 0.25 = 0.75.
Question 19
Two fair coins are tossed simultaneously. What is the exact probability of both coins landing on 'Heads'?
- A) 1/2
- B) 1/4
- C) 3/4
- D) 1/8
Show answer & explanation
Answer: B) 1/4
The total possible outcomes for two coin tosses are (H,H), (H,T), (T,H), and (T,T). Since only one outcome (H,H) is 'both heads' out of the four possibilities, the probability is 1/4.
Question 20
Which of the following is a fundamental mathematical property of any valid individual probability value?
- A) It must be a positive whole number.
- B) It must strictly be greater than 1.
- C) It must always fall between 0 and 1, inclusive.
- D) It can be any real negative number.
Show answer & explanation
Answer: C) It must always fall between 0 and 1, inclusive.
Probability is measured on a scale from 0 (meaning the event is absolutely impossible) to 1 (meaning the event is absolutely certain to occur).
Question 21
Calculate the probability of drawing either a 'King' or a 'Queen' from a standard well-shuffled deck of 52 playing cards.
- A) 1/13
- B) 2/13
- C) 4/13
- D) 1/4
Show answer & explanation
Answer: B) 2/13
There are 4 Kings and 4 Queens in a standard 52-card deck. The total number of successful outcomes is 8. The probability is 8/52, which simplifies to 2/13.
Question 22
In a box of 100 industrial parts, 5 are defective. If one part is chosen at random, what is the probability that it is NOT defective?
- A) 0.05
- B) 0.50
- C) 0.95
- D) 1.00
Show answer & explanation
Answer: C) 0.95
If 5 parts are defective, then 95 parts are good (not defective). The probability of choosing a good part is 95 out of 100, which is exactly 0.95.
Question 23
If events A and B are independent, how is the probability of both occurring (A ∩ B) mathematically calculated?
- A) P(A) + P(B)
- B) P(A) - P(B)
- C) P(A) * P(B)
- D) P(A) / P(B)
Show answer & explanation
Answer: C) P(A) * P(B)
The multiplication rule for independent events states that the probability of both occurring is the product of their individual probabilities.
Question 24
What is the probability of a person winning a prize if the numerical odds 'against' them winning are precisely 7 to 3?
- A) 3/7
- B) 7/10
- C) 3/10
- D) 1/3
Show answer & explanation
Answer: C) 3/10
Odds 'against' of 7:3 mean there are 7 unfavorable outcomes for every 3 favorable outcomes. The total possible outcomes are 7 + 3 = 10. Therefore, the probability of winning is 3/10.
Question 25
In a bag containing 5 red, 3 green, and 2 blue marbles, what is the probability of randomly picking a marble that is NOT red?
- A) 1/2
- B) 3/10
- C) 2/10
- D) 5/10
Show answer & explanation
Answer: A) 1/2
Total marbles = 5 + 3 + 2 = 10. Marbles that are not red (green and blue) = 3 + 2 = 5. The probability of picking one is 5/10, which simplifies to 1/2.
Question 26
Conditional probability P(A|B) refers to:
- A) The probability of A and B happening together.
- B) The probability of A given that B has already occurred.
- C) The total probability of all sample outcomes.
- D) The probability of A multiplied by the probability of B.
Show answer & explanation
Answer: B) The probability of A given that B has already occurred.
The notation P(A|B) is the conditional probability of Event A occurring, given the specific condition that Event B has already happened.
Question 27
Which mathematical rule states that for any two events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B)?
- A) The Product Rule
- B) The Division Rule
- C) The Addition Rule
- D) The Rule of Complements
Show answer & explanation
Answer: C) The Addition Rule
The general Addition Rule for probability is used to find the probability of either event A or B occurring, accounting for the possibility that they could overlap.
Question 28
What is the probability of rolling a total sum of 7 with two fair six-sided dice?
- A) 1/6
- B) 1/12
- C) 1/36
- D) 5/36
Show answer & explanation
Answer: A) 1/6
There are 36 total outcomes (6x6). The outcomes that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Since there are 6 successful outcomes, the probability is 6/36 = 1/6.
Question 29
If you roll a die once, which of the following pairs of events are collectively exhaustive?
- A) Rolling a 2 and rolling a 4.
- B) Rolling an even number and rolling a number less than 3.
- C) Rolling an even number and rolling an odd number.
- D) Rolling a 6 and rolling a 1.
Show answer & explanation
Answer: C) Rolling an even number and rolling an odd number.
Events are collectively exhaustive if their union covers every possible outcome in the sample space. Every number on a die is either even or odd, covering the entire range (1-6).
Question 30
In a Venn diagram, the overlapping section between two circles representing Event A and Event B denotes:
- A) The union of A and B.
- B) The complement of A.
- C) The intersection of A and B.
- D) The exclusion of B from A.
Show answer & explanation
Answer: C) The intersection of A and B.
In set theory and probability, the intersection represents the specific outcomes that belong to both Event A and Event B simultaneously.
Question 31
Which of the following statements regarding the use of a scatter diagram in statistical analysis is most accurate?
- A) It definitively proves causation between two variables.
- B) It visually displays a relationship, but it might indicate a relationship where there fundamentally is none.
- C) It mathematically calculates the exact standard error.
- D) It forces a non-linear relationship to become strictly linear.
Show answer & explanation
Answer: B) It visually displays a relationship, but it might indicate a relationship where there fundamentally is none.
A scatter diagram plots coordinate points to visually show potential trends or correlations. However, statisticians must be cautious, as it may suggest a relationship (spurious correlation) even when no logical or causal link exists.
Question 32
Which of the following correctly defines the mathematical range within which the Pearson correlation coefficient (r) must always lie?
- A) 0 to +1
- B) 0 to +100
- C) -1 to +1
- D) -100 to +100
Show answer & explanation
Answer: C) -1 to +1
The correlation coefficient (r) is an index of linear relationship strength that is mathematically bounded. It can never exceed +1 (perfect positive correlation) and can never drop below -1 (perfect negative correlation).
Question 33
If the correlation coefficient (r) between two economic variables is exactly -0.80, what is the resulting coefficient of determination, and what does it signify?
- A) -0.80; meaning an 80% inverse relationship.
- B) 0.64; meaning 64% of the variation in the dependent variable is explained by the independent variable.
- C) -0.64; meaning a 64% negative variance.
- D) 0.80; meaning an 80% positive variance.
Show answer & explanation
Answer: B) 0.64; meaning 64% of the variation in the dependent variable is explained by the independent variable.
The coefficient of determination is calculated by squaring r. (-0.80)^2 = 0.64. This metric indicates that 64% of the mathematical variance in variable Y can be reliably explained by the linear relationship with variable X.
Question 34
If a researcher adds 10 to every observation of variable X and multiplies every observation of variable Y by 5, what will logically happen to an original correlation coefficient of 0.65?
- A) It will systematically increase.
- B) It will decrease due to the addition of 10.
- C) It will remain completely unchanged at 0.65.
- D) It will become greater than 1.
Show answer & explanation
Answer: C) It will remain completely unchanged at 0.65.
The Pearson correlation coefficient is entirely independent of both the change of origin (adding/subtracting a constant) and the change of scale (multiplying/dividing by a constant).
Question 35
In a formulated regression equation represented as Y = 250 - 4.5X, what does the value '-4.5' specifically represent?
- A) The fixed y-intercept.
- B) The independent variable.
- C) The coefficient of correlation.
- D) The mathematical slope of the regression line.
Show answer & explanation
Answer: D) The mathematical slope of the regression line.
In the standard regression format Y = a + bX, 'b' is the slope. Here, -4.5 is the slope, indicating that for every 1 unit increase in X, the predicted value of Y will mathematically decrease by 4.5 units.
Question 36
Which specific criterion does the 'Method of Least Squares' utilize to determine the optimal line of best fit?
- A) It minimizes the sum of the squared vertical deviations (residuals) between the actual data points and the regression line.
- B) It maximizes the sum of the horizontal deviations.
- C) It ensures the line crosses the y-axis at zero.
- D) It forces all data points to sit directly on the line.
Show answer & explanation
Answer: A) It minimizes the sum of the squared vertical deviations (residuals) between the actual data points and the regression line.
The least squares regression line is mathematically derived by minimizing the total sum of the squares of the vertical errors (the difference between the observed Y values and the predicted Y values).
Question 37
If the correlation coefficient between variables X and Y is exactly 0.00, what visual pattern would their scatter diagram most likely present?
- A) A perfectly straight, horizontal line.
- B) A perfectly straight, vertical line.
- C) A randomly dispersed cloud of points with no discernible linear trend.
- D) A parabolic U-curve.
Show answer & explanation
Answer: C) A randomly dispersed cloud of points with no discernible linear trend.
An 'r' value of 0 mathematically indicates that there is absolutely no linear correlation between the two variables, resulting in a random scatter of data points across the coordinate plane.
Question 38
A regression line predicting total cost (Y) based on units produced (X) is given as Y = 12,000 + 40X. What is the predicted total cost if 500 units are produced?
- A) Rs. 12,000
- B) Rs. 20,000
- C) Rs. 32,000
- D) Rs. 52,000
Show answer & explanation
Answer: C) Rs. 32,000
Substitute X = 500 into the regression equation. Y = 12,000 + 40(500) = 12,000 + 20,000 = Rs. 32,000.
Question 39
Which of the following is a significant mathematical risk when using a linear regression model for 'extrapolation'?
- A) The regression line's slope might instantly invert.
- B) Predicting values entirely outside the historical range assumes the linear relationship continues indefinitely, which may be false.
- C) It automatically turns the correlation coefficient into a negative value.
- D) It forces the y-intercept to become an outlier.
Show answer & explanation
Answer: B) Predicting values entirely outside the historical range assumes the linear relationship continues indefinitely, which may be false.
Extrapolation implies predicting outcomes beyond the limits of the sampled data. This is risky because the established linear relationship might flatten, curve, or break down completely at extreme levels.
Question 40
If variable X and variable Y exhibit a 'perfect positive correlation', what must be the mathematical value of r?
- A) 0
- B) 0.5
- C) 1
- D) 100
Show answer & explanation
Answer: C) 1
A perfect positive correlation signifies that all data points fall exactly on an upward-sloping straight line. In statistical terms, this is strictly represented by an r value of exactly +1.
Question 41
A regression line of Y on X must always mathematically pass through which specific coordinate point on a scatter diagram?
- A) The origin point (0,0).
- B) The absolute highest recorded data point.
- C) The point representing the mean of X and the mean of Y (x-bar, y-bar).
- D) The exact center of the x-axis.
Show answer & explanation
Answer: C) The point representing the mean of X and the mean of Y (x-bar, y-bar).
A foundational property of the least squares regression method is that the calculated line of best fit must strictly intersect the centroid of the dataset, which is the intersection of the variables' means.
Question 42
What does a correlation coefficient (r) of -0.92 mathematically indicate about the relationship between two variables?
- A) A weak positive linear relationship.
- B) A strong positive linear relationship.
- C) A weak negative linear relationship.
- D) A strong negative linear relationship.
Show answer & explanation
Answer: D) A strong negative linear relationship.
The negative sign indicates an inverse relationship (as one variable rises, the other falls). The value 0.92 is very close to 1.0, indicating the linear relationship is exceptionally strong.
Question 43
In a regression analysis testing how rainfall (in mm) affects crop yield (in tons), which variable is considered the 'independent variable'?
- A) Crop yield
- B) Rainfall
- C) The y-intercept
- D) The standard error
Show answer & explanation
Answer: B) Rainfall
The independent variable (X) is the predictor or cause, which in this case is the rainfall. The crop yield is the dependent variable (Y) because its value mathematically depends on the amount of rain.
Question 44
If you calculate the regression line of Y on X, and then subsequently calculate the regression line of X on Y, under what strict condition will the two lines be perfectly identical?
- A) When r = 0
- B) When r = +1 or -1
- C) When the y-intercept is 0
- D) The two lines can never be identical.
Show answer & explanation
Answer: B) When r = +1 or -1
The regression line of Y on X minimizes vertical errors, while X on Y minimizes horizontal errors. These two mathematical lines will only perfectly align if there is a perfect correlation (r = +1 or -1) meaning zero error.
Question 45
If a set of data yields a regression equation of Y = 10 + 0X, what is the value of the correlation coefficient (r)?
- A) -1
- B) 0
- C) 1
- D) Cannot be determined
Show answer & explanation
Answer: B) 0
A slope (b) of 0 mathematically means that changes in X have absolutely no effect on Y, resulting in a flat horizontal line. This strictly implies there is zero linear correlation (r = 0) between the variables.
Question 46
Which of the following correctly defines the exact mathematical range within which the Pearson correlation coefficient (r) must always lie?
- A) 0 to +1
- B) 0 to +100
- C) -1 to +1
- D) -100 to +100
Show answer & explanation
Answer: C) -1 to +1
The correlation coefficient (r) measures the strength and direction of a linear relationship. It is mathematically bounded and can never exceed +1 (perfect positive correlation) nor drop below -1 (perfect negative correlation).
Question 47
If the correlation coefficient (r) between two economic variables is exactly -0.70, what is the resulting 'coefficient of determination', and what does it signify?
- A) -0.70; meaning a 70% inverse relationship.
- B) 0.49; meaning 49% of the variation in the dependent variable is mathematically explained by the independent variable.
- C) -0.49; meaning a 49% negative variance.
- D) 0.70; meaning a 70% positive variance.
Show answer & explanation
Answer: B) 0.49; meaning 49% of the variation in the dependent variable is mathematically explained by the independent variable.
The coefficient of determination is calculated by squaring r. (-0.70)^2 = 0.49. This metric clearly indicates that 49% of the variance in variable Y can be reliably explained by its linear relationship with variable X.
Question 48
If a researcher adds a constant of 12 to every observation of variable X, and divides every observation of variable Y by 3, what will logically happen to an original correlation coefficient of 0.85?
- A) It will systematically increase.
- B) It will decrease due to the division.
- C) It will remain completely unchanged at 0.85.
- D) It will invert to -0.85.
Show answer & explanation
Answer: C) It will remain completely unchanged at 0.85.
A fundamental property of the Pearson correlation coefficient is that it is completely independent of both the change of origin (adding/subtracting a constant) and the change of scale (multiplying/dividing by a positive constant).
Question 49
In a formulated linear regression equation represented as Y = 500 - 6.5X, what does the value '-6.5' specifically represent?
- A) The fixed y-intercept.
- B) The independent variable.
- C) The coefficient of correlation.
- D) The mathematical slope of the regression line.
Show answer & explanation
Answer: D) The mathematical slope of the regression line.
In the standard regression format Y = a + bX, 'b' represents the slope. Here, -6.5 is the slope, indicating that for every 1 unit increase in X, the predicted value of Y mathematically decreases by 6.5 units.
Question 50
If variable X and variable Y exhibit a 'perfect negative correlation', what must be the exact mathematical value of r?
- A) -1
- B) 0
- C) -100
- D) 1
Show answer & explanation
Answer: A) -1
A perfect negative correlation signifies that all data points fall perfectly on a downward-sloping straight line, with zero deviation. In statistical terms, this is strictly represented by an r value of exactly -1.
Question 51
If the calculated correlation coefficient between variables X and Y is exactly 0.00, what visual pattern would their scatter diagram most likely present?
- A) A perfectly straight, horizontal line.
- B) A perfectly straight, upward-sloping line.
- C) A randomly dispersed cloud of points demonstrating absolutely no discernible linear trend.
- D) A parabolic U-curve.
Show answer & explanation
Answer: C) A randomly dispersed cloud of points demonstrating absolutely no discernible linear trend.
An 'r' value of 0 mathematically dictates that there is absolutely no linear correlation between the two variables, resulting in a completely random scatter of data points across the coordinate plane.
Question 52
A regression line of Y on X must mathematically always pass perfectly through which specific coordinate point on a scatter diagram?
- A) The origin point (0,0).
- B) The absolute lowest recorded data point.
- C) The point representing the mean of X and the mean of Y (x̄, ȳ).
- D) The y-intercept of X.
Show answer & explanation
Answer: C) The point representing the mean of X and the mean of Y (x̄, ȳ).
A foundational algebraic property of the least squares regression method dictates that the calculated line of best fit must strictly intersect the centroid of the dataset, which is the exact intersection of the variables' means.
Question 53
Which of the following describes a significant statistical risk when utilizing a linear regression model for 'extrapolation'?
- A) The regression line's slope might instantly invert.
- B) Predicting values entirely outside the historical data range assumes the historical linear relationship continues indefinitely, which may prove violently false.
- C) It automatically turns the correlation coefficient into a negative value.
- D) It forces the y-intercept to become exactly zero.
Show answer & explanation
Answer: B) Predicting values entirely outside the historical data range assumes the historical linear relationship continues indefinitely, which may prove violently false.
Extrapolation implies forecasting outcomes beyond the limits of the sampled data. This is highly risky because the established linear relationship might flatten out, curve, or break down completely at extreme, untested levels.
Question 54
In a regression analysis testing how the amount of fertilizer applied (in kg) affects the total crop yield (in tons), which variable is considered the 'dependent variable'?
- A) The fertilizer applied
- B) The crop yield
- C) The y-intercept
- D) The correlation coefficient
Show answer & explanation
Answer: B) The crop yield
The dependent variable (Y) is the outcome being predicted, which in this case is the crop yield, because its final value mathematically depends entirely on the amount of fertilizer applied (the independent variable, X).
Question 55
Which specific mathematical criterion does the 'Method of Least Squares' utilize to determine the optimal line of best fit?
- A) It maximizes the sum of the horizontal deviations.
- B) It strictly minimizes the sum of the squared vertical deviations (residuals) between the actual data points and the regression line.
- C) It ensures the line crosses the y-axis at zero.
- D) It forces all data points to sit directly on the line.
Show answer & explanation
Answer: B) It strictly minimizes the sum of the squared vertical deviations (residuals) between the actual data points and the regression line.
The least squares regression line is mathematically derived by strictly minimizing the total sum of the squares of the vertical errors (the physical difference between the observed Y values and the regression's predicted Y values).
Question 56
A regression line predicting total factory cost (Y) based on units produced (X) is given exactly as Y = 15,000 + 60X. What is the predicted total cost if 400 units are produced?
- A) Rs. 15,000
- B) Rs. 24,000
- C) Rs. 39,000
- D) Rs. 55,000
Show answer & explanation
Answer: C) Rs. 39,000
Simply substitute X = 400 into the established regression equation. Y = 15,000 + 60(400) = 15,000 + 24,000 = Rs. 39,000.
Question 57
If you calculate the regression line of Y on X, and then subsequently calculate the regression line of X on Y, under what strict mathematical condition will the two lines graphically be perfectly identical?
- A) When r = 0
- B) When r = +1 or -1
- C) When the y-intercept is precisely 0
- D) The two lines can never physically be identical.
Show answer & explanation
Answer: B) When r = +1 or -1
The regression line of Y on X minimizes vertical errors, while X on Y minimizes horizontal errors. These two lines will only perfectly superimpose over each other if there is a perfect correlation (r = +1 or -1), meaning zero scatter error.
Question 58
What is the primary statistical function and inherent risk of utilizing a 'scatter diagram'?
- A) It definitively proves causation between two variables.
- B) It visually displays a relationship, but it carries the risk of indicating a spurious relationship where there fundamentally is no logical connection.
- C) It automatically calculates the variance.
- D) It ensures non-linear relationships become linear.
Show answer & explanation
Answer: B) It visually displays a relationship, but it carries the risk of indicating a spurious relationship where there fundamentally is no logical connection.
A scatter diagram plots points to visually identify potential trends. However, statisticians must be cautious, as it may suggest a strong correlation (a spurious correlation) even when no logical or causal link actually exists between the variables.
Question 59
If a set of data yields a flat regression equation of Y = 25 + 0X, what must be the mathematical value of the correlation coefficient (r)?
- A) -1
- B) 0
- C) 1
- D) Cannot be determined
Show answer & explanation
Answer: B) 0
A calculated slope (b) of exactly 0 mathematically indicates that changes in X have absolutely no predictive effect on Y, resulting in a flat horizontal line. This strictly implies there is zero linear correlation (r = 0).
Question 60
If the slope 'b' in a calculated regression equation is a negative number, what must inherently be true about the correlation coefficient 'r'?
- A) It must be positive.
- B) It must be exactly zero.
- C) It must also be negative.
- D) It must be greater than 1.
Show answer & explanation
Answer: C) It must also be negative.
The mathematical sign of the regression slope (b) and the correlation coefficient (r) must always strictly match. If the line slopes downward (negative b), it inherently reflects an inverse relationship (negative r).