Activities 15 & 16 Elaboration

A linear model is fitted to a data set where the scatter plot shows a linear pattern. The model is then plotted on the scatter plot. Select the best answer

• The line must pass through all of the observed values.

• The line must pass through some of the observed values.

• More of the observed values should be above the line than below the line

• More of the observed values should be below the line than above the line.

• The line must have a slope that follows the same general pattern seen in the scatter plot.

Answer explanation

A trend line is a line added to a scatter plot that shows the generalization of all the data.

Brett has a snow-cone stand outside the local roller-skating rink. He has noticed that his sales tend to increase as the temperature rises. He collected the data below over the last 12 weeks. Which of the following functions provides the best estimate of snow-cone sales for a week with an average temperature of x degrees Fahrenheit?

• f (x) = 1.03x − 32

• f (x) = 1.32x − 52

• f (x) = 0.96x − 31

• f (x) = 0.50x + 10

Six adults took the same standardized test with 30 questions. The age and test result of each adult are shown below. Use a linear model of the data to predict the number of questions that would be answered correctly by a 44-year-old adult.

• 20

• 22

• 24

• 26

Angel looked at data from 100 cities of various populations and found that cities that had more libraries also had more car accidents. Is it reasonable for Angel to conclude that an increase in the number of libraries causes the number of car accidents to increase?

• Yes. There is no way that an increase in the number of car accidents could cause an increase in the number of libraries, so the increase in the number of libraries must have caused the number of car accidents to increase.

• No. There is no association between the number of libraries and car accidents, so there is no way that the increase in libraries could cause the number of car accidents to increase.

• No. Although there is a positive association between the number of libraries and car accidents, there may be other factors that have caused both to increase.

• Yes. There is a positive association between the number of libraries and the number of car accidents, which implies that the increase in the number of libraries caused the number of car accidents to increase.

Mrs. Valladares is interested in knowing whether increased absences from school lower students' GPAs. The data table below shows numbers of absences and GPAs of 10 randomly selected students. Mrs. Valladares concludes that more absences cause a student's GPA to be lower. What is wrong with her reasoning?

• There is no correlation between absences and GPA, so this is evidence that increased absences do not cause lower GPAs.

• There is no correlation between absences and GPA, so there is no evidence that increased absences cause lower GPAs.

• There is a negative linear correlation between absences and GPA, but this is insufficient to determine whether the number of absences is causally related to GPA.

• There is a negative linear correlation between absences and GPA, so instead it is the lower GPA that causes more absences.

Nathan gathered data about the mass, x, and capacity, y, of the drinking cups in the kitchen cabinet. Using a calculator, he created the scatter plot and regression line below from the data. Nathan concluded that high capacity causes cups to be lighter. Which of the following best describes Nathan's conclusion?

• Nathan's conclusion might be true, but there is insufficient evidence to support the conclusion. Correlation does not imply causation.

• Nathan's conclusion is false. There is no correlation in the data, so there is no causal relationship between the mass and capacity of the cups.

• Nathan's conclusion is true. There is a strong negative linear correlation between mass and capacity, which means high capacity causes a cup to be lighter.

• Nathan's conclusion is false. There is a strong negative linear correlation between mass and capacity, which means the heaviness of a cup causes it to hold less.

If 2 x + y ≤ 5 and y ≥ x - 1 which of the following graphs is a correct solution?

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Which following system of inequalities describes the graph below?

• y < 2x

  x + y ≥ 3

• y ≤ 2x

  x + y > 3

• y > 2x

  x + y ≥ 3

• y ≤ 2x

  y > x + 3

Which of the following graphs shows the solution to the following system of linear inequalities?

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Which of these graphs best repesents y ≤ 3 ?

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