# Introduction to Trend Lines - Grade 9 Statistics & Probability ## Section A: Multiple Choice Questions
Q1: A scatter plot shows a set of points where, as the x-values increase, the y-values also increase. What type of correlation does this represent? (a) Positive correlation (b) Negative correlation (c) No correlation (d) Weak correlation
Solution:
Ans: (a) Explanation: When both variables increase together, the relationship shows a positive correlation. Option (b) would occur if y-values decreased as x-values increased. Option (c) would mean no relationship exists. Option (d) describes the strength, not the direction of correlation.
Q2: Which of the following best describes a line of best fit? (a) A line that passes through all data points on a scatter plot (b) A line that approximates the trend of the data with roughly equal numbers of points above and below it (c) A line that only passes through the first and last data points (d) A line that connects all the maximum values in the data set
Solution:
Ans: (b) Explanation: A line of best fit (or trend line) approximates the overall pattern of the data, with points distributed relatively evenly above and below the line. Option (a) is incorrect because data points rarely all fall perfectly on one line. Options (c) and (d) describe arbitrary lines that don't represent the overall trend.
Q3: A scatter plot shows points scattered randomly with no clear pattern. What is the correlation coefficient closest to? (a) \(r = 1\) (b) \(r = -1\) (c) \(r = 0\) (d) \(r = 0.5\)
Solution:
Ans: (c) Explanation: When there is no correlation between variables, the correlation coefficient \(r\) is close to 0. Values of \(r = 1\) or \(r = -1\) indicate perfect positive or negative correlations respectively, while \(r = 0.5\) indicates moderate positive correlation.
Q4: The equation of a trend line is given as \(y = 3x + 5\). What does the slope of 3 represent? (a) The starting value of y when x is zero (b) The rate of change in y for each unit increase in x (c) The maximum value of y (d) The total number of data points
Solution:
Ans: (b) Explanation: In the equation \(y = mx + b\), the slope \(m = 3\) represents the rate of change of y with respect to x. For every 1-unit increase in x, y increases by 3 units. Option (a) describes the y-intercept (5 in this case), while options (c) and (d) are not represented by the slope.
Q5: Which correlation coefficient indicates the strongest relationship between two variables? (a) \(r = 0.3\) (b) \(r = -0.95\) (c) \(r = 0.65\) (d) \(r = -0.15\)
Solution:
Ans: (b) Explanation: The strength of correlation is determined by the absolute value of \(r\). The closer |r| is to 1, the stronger the relationship. Here, \(|-0.95| = 0.95\) is closest to 1, indicating the strongest correlation. The negative sign indicates direction (negative correlation), not strength.
Q6: Using the trend line equation \(y = -2x + 10\), what is the predicted value of y when \(x = 4\)? (a) 2 (b) 18 (c) -8 (d) 6
Solution:
Ans: (a) Explanation: Substitute \(x = 4\) into the equation: \(y = -2(4) + 10\) \(y = -8 + 10\) \(y = 2\) This demonstrates how to use a trend line equation to make predictions.
Q7: A scatter plot shows that as study time increases, test scores increase. The correlation coefficient is \(r = 0.85\). How would you describe this correlation? (a) Weak positive correlation (b) Strong positive correlation (c) Weak negative correlation (d) Strong negative correlation
Solution:
Ans: (b) Explanation: Since \(r = 0.85\) is positive and close to 1, this indicates a strong positive correlation. Values of \(r\) between 0.7 and 1.0 generally indicate strong positive relationships, while values between 0 and 0.3 indicate weak positive correlations.
Q8: In the equation of a trend line \(y = mx + b\), what does the variable \(b\) represent? (a) The slope of the line (b) The y-intercept (c) The x-intercept (d) The correlation coefficient
Solution:
Ans: (b) Explanation: In the slope-intercept form \(y = mx + b\), the constant \(b\) represents the y-intercept, which is the value of y when \(x = 0\). The slope is represented by \(m\), option (c) would require solving for when \(y = 0\), and option (d) is a separate statistical measure.
## Section B: Fill in the Blanks Q9: A __________ is a graph that displays the relationship between two quantitative variables using points plotted on a coordinate plane.
Solution:
Ans: scatter plot Explanation: A scatter plot is the fundamental graphical tool used to visualize the relationship between two variables and determine if a trend exists.
Q10: When two variables show a pattern where one variable increases as the other decreases, this is called a __________ correlation.
Solution:
Ans: negative Explanation: A negative correlation (or inverse relationship) occurs when variables move in opposite directions. This is also called an inverse relationship.
Q11: The __________ is a numerical value between -1 and 1 that measures the strength and direction of a linear relationship between two variables.
Solution:
Ans: correlation coefficient Explanation: The correlation coefficient, denoted as \(r\), quantifies both the strength (how closely points follow a line) and direction (positive or negative) of a linear relationship.
Q12: Using a trend line to estimate values within the range of the data is called __________.
Solution:
Ans: interpolation Explanation:Interpolation involves making predictions for values that fall within the existing data range, which is generally more reliable than extrapolation.
Q13: A trend line drawn on a scatter plot is also known as a line of __________ __________.
Solution:
Ans: best fit Explanation: The line of best fit is the line that best represents the trend in the data, minimizing the overall distance between the line and all data points.
Q14: When using a trend line to predict values outside the range of the original data, this process is called __________.
Solution:
Ans: extrapolation Explanation:Extrapolation extends predictions beyond the data range. This is less reliable than interpolation because it assumes the trend continues unchanged outside the observed range.
## Section C: Word Problems Q15: A biology student records the temperature (in °C) and the number of times a cricket chirps per minute. She creates a trend line with the equation \(y = 4x - 30\), where x represents temperature and y represents chirps per minute. Predict the number of chirps per minute when the temperature is 25°C.
Plot these points mentally and determine if there is a positive, negative, or no correlation. Then describe the relationship in words.
Solution:
Ans: Analysis: As hours studied increase from 2 to 7, test scores increase from 65 to 90. Correlation type: Positive correlation Final Answer: There is a strong positive correlation; as hours studied increase, test scores increase consistently.
Q17: A car dealership tracks the age of used cars (in years) and their selling prices (in thousands of dollars). The trend line equation is \(y = -1.5x + 22\), where x is the age in years and y is the price in thousands. What is the predicted selling price of a car that is 8 years old?
Q18: A fitness tracker shows that a person burns calories based on minutes of exercise. The trend line equation is \(y = 6x + 50\), where x is minutes of exercise and y is total calories burned. If someone wants to burn 200 calories, approximately how many minutes should they exercise?
Solution:
Ans: Step 1: Write the equation: \(y = 6x + 50\) Step 2: Substitute \(y = 200\): \(200 = 6x + 50\) Step 3: Subtract 50 from both sides: \(150 = 6x\) Step 4: Divide by 6: \(x = 25\) Final Answer: 25 minutes
Q19: A scatter plot displays the relationship between the number of hours per week students play video games and their GPA. The correlation coefficient is calculated to be \(r = -0.78\). Interpret this value by describing the strength and direction of the correlation.
Solution:
Ans: Direction: Negative (as video game hours increase, GPA tends to decrease) Strength: Strong (|r| = 0.78 is close to 1) Final Answer: There is a strong negative correlation between hours playing video games and GPA, meaning students who play more video games tend to have lower GPAs.
Q20: A coffee shop records the outdoor temperature (in °F) and the number of hot coffee drinks sold. The trend line equation is \(y = -3x + 270\), where x is temperature and y is number of drinks sold. Using this equation, find the temperature at which the shop would predict selling 150 hot coffees.
Solution:
Ans: Step 1: Write the equation: \(y = -3x + 270\) Step 2: Substitute \(y = 150\): \(150 = -3x + 270\) Step 3: Subtract 270 from both sides: \(-120 = -3x\) Step 4: Divide by -3: \(x = 40\) Final Answer: 40°F
The document Worksheet (with Solutions): Introduction To Trend Lines is a part of the Grade 9 Course Statistics & Probability.
Semester Notes, Worksheet (with Solutions): Introduction To Trend Lines, Exam, Previous Year Questions with Solutions, pdf , past year papers, Summary, study material, Viva Questions, shortcuts and tricks, Worksheet (with Solutions): Introduction To Trend Lines, Important questions, Objective type Questions, Extra Questions, Worksheet (with Solutions): Introduction To Trend Lines, ppt, MCQs, practice quizzes, video lectures, mock tests for examination, Free, Sample Paper;
