Q1: A scatter plot shows the relationship between hours studied and test scores. The least-squares regression line is used to: (a) Maximize the sum of the residuals (b) Minimize the sum of the squared residuals (c) Maximize the correlation coefficient (d) Make all residuals equal to zero
Solution:
Ans: (b) Explanation: The least-squares regression line minimizes the sum of the squared residuals. A residual is the difference between an observed value and the predicted value. Squaring ensures that positive and negative differences don't cancel out, and minimizing this sum gives the best-fit line.
Q2: The equation of a least-squares regression line is written as \(\hat{y} = a + bx\). What does the symbol \(\hat{y}\) represent? (a) The actual observed value of the response variable (b) The predicted value of the response variable (c) The explanatory variable (d) The residual value
Solution:
Ans: (b) Explanation: The symbol \(\hat{y}\) (read as "y-hat") represents the predicted value of the response variable based on the regression equation. The actual observed value is denoted simply as \(y\), and the difference \(y - \hat{y}\) is the residual.
Q3: In the regression equation \(\hat{y} = 5 + 3x\), what is the slope of the line? (a) 5 (b) 3 (c) 8 (d) 15
Solution:
Ans: (b) Explanation: In the equation \(\hat{y} = a + bx\), the slope is represented by \(b\), the coefficient of \(x\). Here, \(b = 3\), which means for every one-unit increase in \(x\), \(\hat{y}\) increases by 3 units. The value 5 is the y-intercept.
Q4: For a regression line \(\hat{y} = 12 - 2x\), what does the y-intercept represent? (a) The predicted value of \(y\) when \(x = 1\) (b) The predicted value of \(y\) when \(x = 0\) (c) The slope of the line (d) The average residual
Solution:
Ans: (b) Explanation: The y-intercept is the value of \(\hat{y}\) when \(x = 0\). In this equation, when \(x = 0\), \(\hat{y} = 12 - 2(0) = 12\). The y-intercept represents the starting value of the response variable when the explanatory variable is zero.
Ans: (b) Explanation: A residual is the difference between the observed value \(y\) and the predicted value \(\hat{y}\). It is calculated as \(y - \hat{y}\). Positive residuals indicate the actual value is above the regression line, while negative residuals indicate it is below.
Q6: The correlation coefficient \(r = 0.85\) between two variables. What does this indicate about the relationship? (a) Strong negative linear relationship (b) Weak positive linear relationship (c) Strong positive linear relationship (d) No linear relationship
Solution:
Ans: (c) Explanation: The correlation coefficient \(r\) ranges from -1 to 1. A value of \(r = 0.85\) is close to 1, indicating a strong positive linear relationship between the two variables. Values near -1 indicate strong negative relationships, and values near 0 indicate weak or no linear relationship.
Q7: The formula for the slope \(b\) of the least-squares regression line is \(b = r\frac{s_y}{s_x}\). What do \(s_x\) and \(s_y\) represent? (a) The means of \(x\) and \(y\) (b) The standard deviations of \(x\) and \(y\) (c) The variances of \(x\) and \(y\) (d) The medians of \(x\) and \(y\)
Solution:
Ans: (b) Explanation: In the slope formula \(b = r\frac{s_y}{s_x}\), \(s_x\) and \(s_y\) represent the standard deviations of the explanatory variable \(x\) and the response variable \(y\), respectively. The correlation coefficient \(r\) is also part of this formula.
Q8: A residual plot shows a random scatter of points around the horizontal axis at zero. This suggests: (a) The linear model is not appropriate (b) The linear model is appropriate (c) There are outliers in the data (d) The correlation is negative
Solution:
Ans: (b) Explanation: A residual plot with a random scatter of points around zero indicates that the linear model is appropriate. If the residual plot shows a clear pattern (such as a curve), it suggests that a linear model may not be the best fit for the data.
Section B: Fill in the Blanks
Q9: The least-squares regression line always passes through the point \((\bar{x}, \bar{y})\), where \(\bar{x}\) and \(\bar{y}\) are the __________ of the \(x\) and \(y\) values.
Solution:
Ans: means Explanation: The least-squares regression line always passes through the point of means, denoted as \((\bar{x}, \bar{y})\). This is a fundamental property of the regression line.
Q10: In a regression equation \(\hat{y} = a + bx\), the coefficient \(a\) is called the __________ and represents the predicted value of \(y\) when \(x = 0\).
Solution:
Ans: y-intercept Explanation: The coefficient \(a\) is the y-intercept of the regression line. It represents the value of \(\hat{y}\) when the explanatory variable \(x\) equals zero.
Q11: The process of using a regression line to predict values within the range of the data is called __________, while predicting beyond the range is called extrapolation.
Solution:
Ans: interpolation Explanation:Interpolation refers to making predictions within the range of observed data, which is generally reliable. Extrapolation means predicting beyond the data range, which can be less accurate.
Q12: The value \(r^2\), called the coefficient of __________, represents the proportion of variance in the response variable that is explained by the regression line.
Solution:
Ans: determination Explanation: The coefficient of determination, denoted \(r^2\), measures how well the regression line fits the data. It represents the proportion of total variation in \(y\) that is explained by the linear relationship with \(x\).
Q13: If the slope of a regression line is negative, it indicates that as the explanatory variable increases, the response variable __________.
Solution:
Ans: decreases Explanation: A negative slope means there is an inverse relationship: as \(x\) increases, \(y\) decreases. Conversely, a positive slope indicates that \(y\) increases as \(x\) increases.
Q14: The sum of all residuals in a least-squares regression always equals __________.
Solution:
Ans: zero (or 0) Explanation: A key property of the least-squares regression line is that the sum of all residuals equals zero. This means the positive and negative residuals balance out.
Section C: Word Problems
Q15: A teacher collects data on the number of hours students study for a test and their scores. The least-squares regression equation is found to be \(\hat{y} = 45 + 8x\), where \(x\) is the number of hours studied and \(\hat{y}\) is the predicted test score. Predict the test score for a student who studies for 5 hours.
Solution:
Ans: Using the regression equation \(\hat{y} = 45 + 8x\): Substitute \(x = 5\): \(\hat{y} = 45 + 8(5)\) \(\hat{y} = 45 + 40\) \(\hat{y} = 85\) Final Answer: 85
Q16: A botanist studies the relationship between the amount of fertilizer (in grams) and plant height (in cm). The regression equation is \(\hat{y} = 12 + 0.5x\). If a plant receives 20 grams of fertilizer, what is the predicted height? Calculate the residual if the actual height is 24 cm.
Step 2: Calculate residual using \(\text{residual} = y - \hat{y}\): \(\text{residual} = 24 - 22\) \(\text{residual} = 2\) cm Final Answer: Predicted height = 22 cm; Residual = 2 cm
Q17: A linear regression analysis of temperature (in °F) and ice cream sales (in dollars) yields the equation \(\hat{y} = -100 + 5x\). What is the y-intercept, and does it have a practical interpretation in this context?
Solution:
Ans: From the equation \(\hat{y} = -100 + 5x\), the y-intercept is \(a = -100\).
The y-intercept represents predicted sales when temperature is 0°F. A predicted sales value of -$100 is not practically meaningful, as sales cannot be negative. This illustrates that the y-intercept may not always have a practical interpretation, especially when \(x = 0\) is outside the range of observed data. Final Answer: The y-intercept is -100, but it has no practical interpretation in this context.
Q18: Given the following statistics for a data set: \(\bar{x} = 10\), \(\bar{y} = 50\), \(s_x = 2\), \(s_y = 8\), and \(r = 0.75\). Calculate the slope \(b\) of the least-squares regression line using the formula \(b = r\frac{s_y}{s_x}\).
Solution:
Ans: Using the formula \(b = r\frac{s_y}{s_x}\): \(b = 0.75 \times \frac{8}{2}\) \(b = 0.75 \times 4\) \(b = 3\) Final Answer: The slope \(b = 3\)
Q19: Using the slope from Q18 (\(b = 3\)) and the point \((\bar{x}, \bar{y}) = (10, 50)\), find the y-intercept \(a\) using the formula \(a = \bar{y} - b\bar{x}\), and write the complete regression equation.
Solution:
Ans: Using the formula \(a = \bar{y} - b\bar{x}\): \(a = 50 - 3(10)\) \(a = 50 - 30\) \(a = 20\)
The regression equation is: \(\hat{y} = a + bx\) \(\hat{y} = 20 + 3x\) Final Answer: \(a = 20\); Regression equation: \(\hat{y} = 20 + 3x\)
Q20: A regression analysis shows that \(r^2 = 0.64\) for the relationship between hours of practice and basketball free-throw percentage. What percentage of the variation in free-throw percentage is explained by hours of practice? What is the correlation coefficient \(r\)?
Solution:
Ans: The coefficient of determination \(r^2 = 0.64\) means that 64% of the variation in free-throw percentage is explained by hours of practice.
To find \(r\), take the square root of \(r^2\): \(r = \pm\sqrt{0.64}\) \(r = \pm 0.8\)
Since the context suggests a positive relationship (more practice improves performance), \(r = 0.8\). Final Answer: 64% of variation is explained; \(r = 0.8\) (assuming positive relationship)
The document Worksheet (with Solutions): Least-Squares Regression Equations is a part of the Grade 9 Course Statistics & Probability.
Objective type Questions, mock tests for examination, Sample Paper, Worksheet (with Solutions): Least-Squares Regression Equations, Worksheet (with Solutions): Least-Squares Regression Equations, Summary, ppt, Viva Questions, Important questions, past year papers, Semester Notes, pdf , Free, study material, Worksheet (with Solutions): Least-Squares Regression Equations, video lectures, Previous Year Questions with Solutions, practice quizzes, MCQs, Exam, shortcuts and tricks, Extra Questions;
