Study Island Copyright 2014 Edmentum - All rights reserved. 1. The table below shows the speed of a ball as it rolls down a slope. 3. David wondered if the temperature outside affects his gas He recorded the temperature and his gas mileage over the last few months. His data is shown below. Roll Time in seconds, x Speed of Rolling Ball 1 2 3 4 5 Speed in inches/second, S 1.2 6.24 20.35 59.87 170.5 Which equation best models this set of data? S = 3.3 x - 2.1 S = 2.8x - 1.6 S = 2.8 x - 1.6 S = 5.04x - 3.84 2. Ten people were chosen at random and surveyed. The survey asked participants for the number of hours they sleep per night and the amount of their annual income. Letting X represent the number of hours the participant sleeps per night and Y represent the participant's annual income, the surveyor calculated the correlation coefficient between X and Y to be 0.37. Interpret the correlation coefficient calculated by choosing the statement below which correctly describes the correlation between X and Y. weak positive correlation strong negative correlation weak negative correlation strong positive correlation What can he conclude from his data? The hotter the temperature, the better his gas There is no relationship between the temperature and his gas The cooler the temperature, the better his gas The hotter the temperature, the worse his gas 4. Concert tickets for a popular band went on sale at 2:00 on Monday. The concert venue has room for 15,000 fans to attend the concert. The number of seats available for the first four hours after tickets went on sale are shown in the table. Concert Tickets Available 4:00 Time on Monday 2:00 3:00 Number of Seats Available 5:00 6:00 15,000 9,000 5,400 3,240 1,944 Which of the following functions would best model the data above? exponential function linear function quadratic function constant function
4. An ice cream company has discovered that the number of ice cream cones sold in one hour is related to the temperature outside. A collection of data from two different stores on the outside temperature and number of cones sold produced the dot plots below. 5. Tony, the lifeguard at Waununa Beach, recorded the temperature and the number of people on his stretch of beach over the course of several weeks. He collected data at noon on Tuesdays and Saturdays during the five weeks between Mother's Day and Father's Day, as shown in the graph below. Without finding the equations, compare the regression models for Store A's data and Store B's data. The models will have different slopes and different intercepts because of an influential point. Store A's slope will be less than Store B's slope. The models will have the same slope but different intercepts because of an influential point. The models will have the same intercepts but different slopes because of an influential point. Store A's slope will be less than Store B's slope. The models will be the same because the data points are the same. Which of the following is a valid conclusion? The increase in temperature caused an increase in the number of people at the beach. No conclusion can be drawn about the correlation or causation of the temperature and the number of people at the beach. The increase in the number of people at the beach is correlated to, but not caused by, the increase in temperature. There is no correlation between the temperature and the number of people at the beach.
7. The graph below shows a line of best fit for data collected on the cost of a cruise as a function of the number of extra activities added to the cruise. 9. The graph below shows a line of best fit for data collected on the daily profits made by street vendors in relation to the number of customers. The equation for the line of best fit is y = 125x + 750. What does the slope of this line represent? the cost per activity the number of activities added the cost of the cruise without any activities the overall cost of the cruise 8. The graph below shows a line of best fit for data collected on the distance drivers traveled as a function of time. The equation of the line of best fit is y = 12 / 5 x - 30. What does the y-intercept of the line represent? the amount street vendors charge per customer the amount it costs a street vendor to operate his or her cart the number of customers needed to earn a profit the total number of customers 10. Berlin, Germany, and El Calafate, Argentina, are about the same distance from the equator. The graph below shows a line of best fit for data collected on the average high temperature in El Calafate as a function of the average high temperature in Berlin. Which of the following is the equation of the line of best fit? Which of the following is the equation of the line of best fit?
Answers 1. C 2. A 3. A 4. A 5. C 6. A 7. A 8. A 9. B 10. C Explanations 1. To determine the type of function needed, look at the differences between the values in the second row (the first differences) and also look at the differences between the first differences (the second differences). First Difference Second Difference 6.24-1.2 = 5.04 14.11-5.04 = 9.07 20.35-6.24 = 14.11 39.52-14.11 = 25.41 59.87-20.35 = 39.52 110.63 - = 39.52 71.11 170.5 - = 59.87 110.63 The first difference values are not constant, so the equation that best models the data will not be linear. The second difference values are not constant, so the equation that best models the data will not be quadratic. Substitute in numbers from the table to see which of the exponential equations fits the data. S = 3.3 x - 2.1 S(1) = 3.3 1-2.1 = 1.2 S(2) = 3.3 2-2.1 = 8.79 This equation does not fit the data, so try the remaining exponential equation. S = 2.8 x - 1.6 S(1) = 2.8 1-1.6 = 1.2 S(2) = 2.8 2-1.6 = 6.24 This equation fits the data, so the equation S = 2.8 x - 1.6 best models this set of data. 2. Correlation coefficients which are close to positive 1 show a strong positive correlation between two variables, while correlation coefficients close to negative 1 show a strong negative correlation. The correlation coefficient is positive and not close to 1. Therefore, there is a weak positive correlation. 3. The temperature is shown on the x-axis, and the gas mileage is shown on the y-axis. As the temperature outside increases, the gas mileage on his car generally increases as well. (High gas mileage is preferred to low gas ) Therefore, David can conclude from his data that the hotter the temperature, the better the gas 4. An influential point is a single point that strongly influences the slope and the intercept of a regression model. If an influential point is removed from the data, the data will yield a very different model. The data points for Store A and Store B are the same except for one point. Store B has an additional data point at 100. The data point at 100 is called an influential point; therefore, it will cause the model for Store B to differ from the model for Store In this case, the influential point causes the model of Store B to have a greater slope than the model of Store The models will have different slopes and
different intercepts because of an influential point. Store A's slope will be less than Store B's slope. 5. Causation between variables indicates that one of the variables causes the other variable. The temperature might be one reason for an increased number of people, but, based on the information given, it cannot be determined to be the sole cause. Other factors might contribute, like the fact that school is over, more people are vacationing, etc. However, Tony could conclude that more people went to the beach as the temperature increased. Therefore, the increase in the number of people at the beach is correlated to, but not caused by, the increase in temperature. 6. To determine the type of function needed, look at the differences between the values in the second row (the first differences) and also look at the differences between the first differences (the second differences). First Difference 9,000 - = - 15,000 6,000 5,400-9,000 = - 3,600 3,240-5,400 = - 2,160 1,944-3,240 = - 1,296 Second Difference -3,600 - (- = 6,000) 2,400-2,160 - (- = 3,600) 1,440-1,296 - (- 2,160) = 864 The first difference values are not constant, so the equation that best models the data will not be linear. The second difference values are not constant, so the equation that best models the data will not be quadratic. Determine if the terms have a common ratio by dividing each term by the previous term. 9,000 15,000 = 0.6 5,400 9,000 = 0.6 3,240 5,400 = 0.6 1,944 3,240 = 0.6 Since the terms have a common ratio, an exponential function best models this set of data. 7. The slope of a line is the change in the quantity on the y-axis per unit of the quantity on the x-axis. In this case, the quantity on the y-axis is the cost in dollars, and the quantity on the x-axis is the number of extra activities. Therefore, the slope represents the cost per activity. 8. The equation of a line can be written as y = mx + b, where m is the slope and b is the y-intercept. The points (3,50) and (6,100) both lie on the line of best fit. Use these points to find the slope of the line. The y-intercept of the line is 0. Therefore, the equation of the line of best fit is shown below. 9. The y-intercept is the value of the quantity on the y-axis when the quantity on the x-axis is zero. In this case, the quantity on the x-axis is the number of customers. When this is zero, a street vendor has not made any money, but he or she still has to cover the expenses required to operate his or her cart. Therefore, the y-intercept is the amount it costs a street vendor to operate his or her cart. 10. The equation of a line can be written as y = mx + b, where m is the slope and b is the y-intercept.
The points (22.5,75) and (75,45) both lie on the line of best fit. Use these points to find the slope of the line. To find the equation of the line of best fit, substitute the slope and the point (75,45) into the point-slope equation, and solve for y.