Q1: Which of the following best describes a scatterplot? (a) A graph that shows the relationship between two quantitative variables (b) A graph that shows only categorical data (c) A graph that displays data using bars of different heights (d) A graph that shows data divided into sectors
Solution:
Ans: (a) Explanation: A scatterplot is a graph that displays the relationship between two quantitative variables by plotting points on a coordinate plane. Option (b) is incorrect because scatterplots use quantitative, not categorical data. Option (c) describes a bar graph, and option (d) describes a pie chart.
Q2: In a scatterplot, if the points generally rise from left to right, what type of association does this indicate? (a) Negative association (b) Positive association (c) No association (d) Constant association
Solution:
Ans: (b) Explanation: When points in a scatterplot rise from left to right, this indicates a positive association between the variables. As one variable increases, the other variable also tends to increase. A negative association would show points falling from left to right, while no association would show no clear pattern.
Q3: A scatterplot shows the relationship between hours studied and test scores. The points form a pattern that decreases from left to right. Which statement is true? (a) As hours studied increase, test scores tend to increase (b) As hours studied increase, test scores tend to decrease (c) There is no relationship between hours studied and test scores (d) Hours studied and test scores are unrelated variables
Solution:
Ans: (b) Explanation: When a scatterplot shows points decreasing from left to right, this indicates a negative association. This means as one variable (hours studied) increases, the other variable (test scores) tends to decrease. However, this scenario is unusual in real life, as typically more study hours would correlate with higher scores.
Q4: Which correlation coefficient indicates the strongest linear relationship? (a) \(r = 0.45\) (b) \(r = -0.89\) (c) \(r = 0.12\) (d) \(r = -0.35\)
Solution:
Ans: (b) Explanation: The correlation coefficient \(r\) measures the strength and direction of a linear relationship. The strength is determined by the absolute value of \(r\). Since \(|-0.89| = 0.89\) is closest to 1, this indicates the strongest relationship. The negative sign indicates direction (negative association), not weakness.
Q5: What does an outlier in a scatterplot represent? (a) A point that follows the general pattern of the data (b) A point that falls far from the general pattern of the other points (c) The average of all data points (d) The center point of the scatterplot
Solution:
Ans: (b) Explanation: An outlier is a data point that falls far from the general pattern formed by the other points in the scatterplot. Outliers can significantly affect the correlation coefficient and the line of best fit. They may represent unusual observations or data entry errors.
Q6: A scatterplot shows no clear pattern, with points scattered randomly. What is the approximate value of the correlation coefficient? (a) \(r = 1\) (b) \(r = -1\) (c) \(r = 0\) (d) \(r = 0.95\)
Solution:
Ans: (c) Explanation: When points are scattered randomly with no clear linear pattern, this indicates no association between the variables. The correlation coefficient \(r\) would be close to 0. Values of \(r = 1\) or \(r = -1\) indicate perfect positive or negative linear relationships, respectively.
Q7: Which of the following values of \(r\) indicates a moderate positive correlation? (a) \(r = -0.65\) (b) \(r = 0.98\) (c) \(r = 0.55\) (d) \(r = 0.05\)
Solution:
Ans: (c) Explanation: A moderate positive correlation is typically indicated by a correlation coefficient between 0.4 and 0.7. The value \(r = 0.55\) falls in this range. Option (a) shows moderate negative correlation, option (b) shows strong positive correlation, and option (d) shows weak or no correlation.
Q8: The line of best fit for a scatterplot has the equation \(y = 3x + 5\). What does the slope of 3 represent? (a) The y-intercept of the line (b) For every 1-unit increase in \(x\), \(y\) increases by 3 units (c) For every 3-unit increase in \(x\), \(y\) increases by 1 unit (d) The starting value of \(y\) when \(x = 0\)
Solution:
Ans: (b) Explanation: In the equation \(y = 3x + 5\), the slope is 3. The slope represents the rate of change: for every 1-unit increase in \(x\), the value of \(y\) increases by 3 units. The y-intercept is 5, which is the value of \(y\) when \(x = 0\).
## Section B: Fill in the Blanks Q9:A graph that shows the relationship between two quantitative variables using plotted points is called a __________.
Solution:
Ans: scatterplot Explanation: A scatterplot is the specific type of graph used to display the relationship between two quantitative variables by plotting ordered pairs as points on a coordinate plane.
Q10:The measure that describes both the strength and direction of a linear relationship between two variables is called the __________.
Solution:
Ans: correlation coefficient Explanation: The correlation coefficient, denoted by \(r\), is a numerical value between -1 and 1 that measures both the strength and direction of a linear relationship between two quantitative variables.
Q11:When two variables both increase together, they have a __________ association.
Solution:
Ans: positive Explanation: A positive association (or positive correlation) occurs when both variables tend to increase together or decrease together. In a scatterplot, this appears as points rising from left to right.
Q12:A straight line that best represents the data on a scatterplot is called the __________.
Solution:
Ans: line of best fit (or trend line) Explanation: The line of best fit, also called the trend line or regression line, is the straight line that comes closest to the data points on a scatterplot and can be used to make predictions.
Q13:The correlation coefficient \(r\) always has a value between __________ and __________.
Solution:
Ans: -1 and 1 Explanation: The correlation coefficient \(r\) ranges from -1 to 1. A value of -1 indicates a perfect negative linear relationship, 1 indicates a perfect positive linear relationship, and 0 indicates no linear relationship.
Q14:In the equation of a line \(y = mx + b\), the letter \(m\) represents the __________ of the line.
Solution:
Ans: slope Explanation: In the slope-intercept form of a linear equation \(y = mx + b\), the coefficient \(m\) represents the slope of the line, which indicates the rate of change of \(y\) with respect to \(x\).
## Section C: Word Problems Q15:A teacher creates a scatterplot showing the relationship between the number of hours students spent on homework (x-axis) and their quiz scores (y-axis). The scatterplot shows points that generally rise from left to right. Describe the association between hours spent on homework and quiz scores, and explain what this means in context.
Solution:
Ans: Final Answer: The scatterplot shows a positive association between hours spent on homework and quiz scores. This means that as students spend more hours on homework, their quiz scores tend to increase. In other words, students who dedicate more time to homework generally perform better on quizzes.
Q16:A scatterplot displays the relationship between the age of a car (in years) and its resale value (in thousands of dollars). The correlation coefficient is \(r = -0.82\). Interpret this correlation coefficient in the context of the problem.
Solution:
Ans: Final Answer: The correlation coefficient of \(r = -0.82\) indicates a strong negative association between the age of a car and its resale value. This means that as the car gets older, its resale value tends to decrease. The value of -0.82 shows this is a strong relationship, meaning age is a good predictor of resale value.
Q17:A scatterplot shows the relationship between daily temperature (in degrees Fahrenheit) and hot chocolate sales at a café. One data point shows that on a day when it was 85°F, the café sold 45 cups of hot chocolate, while on most warm days above 75°F, sales were between 5 and 15 cups. Is this point an outlier? Explain your reasoning.
Solution:
Ans: Final Answer: Yes, this point is an outlier. An outlier is a data point that falls far from the general pattern of the other points. Since most warm days above 75°F had sales between 5 and 15 cups, but this particular 85°F day had sales of 45 cups, this point does not fit the pattern and is significantly higher than expected. There may have been a special event or other unusual circumstance on that day.
Q18:The line of best fit for a scatterplot showing the relationship between hours studied (\(x\)) and test scores (\(y\)) is given by the equation \(y = 4x + 62\). Use this equation to predict the test score for a student who studied for 7 hours.
Solution:
Ans: Step 1: Substitute \(x = 7\) into the equation \(y = 4(7) + 62\) Step 2: Simplify \(y = 28 + 62\) \(y = 90\) Final Answer: The predicted test score is 90 points.
Q19:A fitness center creates a scatterplot to show the relationship between the number of months a person has been a member (\(x\)) and the number of pounds they have lost (\(y\)). The line of best fit has the equation \(y = 2.5x + 3\). What does the y-intercept of 3 mean in this context?
Solution:
Ans: Final Answer: The y-intercept of 3 means that when \(x = 0\) (when a person first joins the fitness center, before any months of membership), the model predicts they will have lost 3 pounds. This could represent initial weight loss from the signup process, orientation activities, or baseline measurement variations. In practical terms, it represents the starting point of the relationship.
Q20:A scatterplot shows the relationship between the number of hours per week students spend playing video games and their grade point average (GPA). Two correlation coefficients are calculated: one including all data points gives \(r = -0.45\), and another excluding one outlier gives \(r = -0.68\). Explain how the outlier affected the correlation coefficient and what this tells us about the relationship between video game time and GPA.
Solution:
Ans: Final Answer: The outlier weakened the correlation coefficient. When the outlier was included, \(r = -0.45\) indicated a moderate negative association. When it was removed, \(r = -0.68\) showed a stronger negative association. This tells us that the outlier did not fit the general pattern of the data. Without the outlier, there is a stronger negative relationship between video game time and GPA, meaning that as students spend more time playing video games, their GPA tends to decrease more consistently. The outlier may have represented a student who plays many video games but still maintains a high GPA, or vice versa.
The document Worksheet (with Solutions): Scatterplots is a part of the Grade 9 Course Integrated Math 1.
Worksheet (with Solutions): Scatterplots, past year papers, Semester Notes, Worksheet (with Solutions): Scatterplots, Extra Questions, Summary, mock tests for examination, Previous Year Questions with Solutions, pdf , shortcuts and tricks, Important questions, MCQs, Exam, Objective type Questions, study material, Worksheet (with Solutions): Scatterplots, Sample Paper, Viva Questions, video lectures, ppt, practice quizzes, Free;
