Unit 2 - Two Variable Data

The least-squares regression model yˆ=−3.4+5.2x and correlation coefficient r=0.66 were calculated for a set of bivariate data with variables x and y. Which of the following is closest to the proportion of the variation in y that cannot be explained by the explanatory variable? 81% 66% 56% 44% 34%

56%

A set of bivariate data was used to create a least-squares regression line. Which of the following is minimized by the line? The sum of the residuals The sum of the squared residuals The sum of the absolute values of the residuals The influence of outliers The slope

The sum of the squared residuals

A real estate agent wants to predict the selling price of single-family homes from the size of each house. A scatterplot created from a sample of houses shows an exponential relationship between price, in thousands of dollars, and size, in 100 square feet. To create a linear model, the natural logarithm of price was taken and the least-squares regression line was given as ln(priceˆ)=2.08+0.11(size). Based on the model, which of the following is closest to the predicted selling price for a house with a size of 3,200 square feet? $54,500 $270,000 $354,000 $398,000 $560,000

$270,000

Bankers at a large financial institution created the linear regression model dˆ=0.37−0.0004s to predict the proportion of customers who would default on their loans, d, based on the customer's credit score, s. For a customer with a credit score of 700, which of the following is true? The default proportion is predicted to be 0.09. The default proportion will be 0.09. The default proportion is predicted to be approximately 1.75 million. The default proportion will be approximately 1.75 million. The default proportion is predicted to be 0.28.

The default proportion is predicted to be 0.09.

A grocery store wants to examine the relationship between the sales amounts each day at two different locations, store A and store B. The sales amount each day, in dollars, was recorded for 10 days at each store. The least-squares regression line is yˆ=−3,000+1.2x, where x represents the sales amounts each day at store A and y represents the sales amounts each day at store B. If the mean of the 10 sales amounts for store B is $45,000, what is the mean of the 10 sales amounts for store A? $35,000 $40,000 $42,000 $45,000 $51,000

$40,000

For a random sample of 20 professional athletes, there is a strong, linear relationship between the number of hours they exercise per week and their resting heart rate. For the athletes in the sample, those who exercise more hours per week tend to have lower resting heart rates than those who exercise less. Which of the following is a reasonable value for the correlation between the number of hours athletes exercise per week and their resting heart rate? 0.71 0.00 −0.14 −0.87 −1.00

-0.87

Which of the following is the best description of a positive association between two variables? The values will create a line when graphed on a scatterplot. The values will create a line with positive slope when graphed on a scatterplot. As the value of one of the variables increases, the value of the other variable tends to decrease. As the value of one of the variables increases, the value of the other variable tends to increase. All values of both variables are positive.

As the value of one of the variables increases, the value of the other variable tends to increase.

A certain cell phone plan charges a fee of $1 for each international call made plus $0.02 for each second of talk time for the international call. A business owner tracked the time and cost for each of the calls made by the employees when they traveled internationally for business. What is the appropriate value of the correlation between time and cost for the international calls? The appropriate value is 1.02 because each call takes at least one second. The appropriate value is 1 because there is a perfect linear relationship between the time of the call and how much it costs. The appropriate value is 0.97 because there is a strong positive relationship between the time and cost of a call. The appropriate value is −0.50−0.50 because the phone company should discount the price for business owners who make many international calls. The appropriate value is 0 because there is no variability in the cost of the calls.

The appropriate value is 1 because there is a perfect linear relationship between the time of the call and how much it costs.

A teacher collected information from a class of 25 students about the time, in hours, they spent studying the previous week and the time, in hours, they spent on the Internet the previous week. The value of the correlation coefficient between hours spent studying and hours spent on the Internet was −0.72. If the teacher changes the units of each variable from hours to minutes, what will be the value of the correlation coefficient between minutes studying and minutes spent on the Internet? −43.2 −0.72 −0.012 0.72 60.72

-0.72

A botanist found a correlation between the length of an aspen leaf and its surface area to be 0.94. Why does the correlation value of 0.94 not necessarily indicate that a linear model is the most appropriate model for the relationship between length of an aspen leaf and its surface area? The value must be exactly 1 or −1−1 to indicate a linear model is the most appropriate model. The value must be 0 to indicate a linear model is the most appropriate model. A causal relationship should be established first before determining the most appropriate model. The value of 0.94 implies that only 88% of the data have a linear relationship. Even with a correlation value of 0.94, it is possible that the relationship could still be better represented by a nonlinear model.

Even with a correlation value of 0.94, it is possible that the relationship could still be better represented by a nonlinear model.

For a specific species of fish in a pond, a wildlife biologist wants to build a regression equation to predict the weight of a fish based on its length. The biologist collects a random sample of this species of fish and finds that the lengths vary from 0.75 to 1.35 inches. The biologist uses the data from the sample to create a single linear regression model. Would it be appropriate to use this model to predict the weight of a fish of this species that is 3 inches long? Yes, because 3 inches falls above the maximum value of lengths in the sample. Yes, because the regression equation is based on a random sample. Yes, because the association between length and weight is positive. No, because 3 inches falls above the maximum value of lengths in the sample. No, because there may not be any 3-inch fish of this species in the pond.

No, because 3 inches falls above the maximum value of lengths in the sample.

At a large airport, data were recorded for one month on how many baggage items were unloaded from each flight upon arrival as well as the time required to deliver all the baggage items on the flight to the baggage claim area. A scatterplot of the two variables indicated a strong, positive linear association between the variables. Which of the following statements is a correct interpretation of the word "strong" in the description of the association? A least-squares model predicts that the more baggage items that are unloaded from a flight, the greater the time required to deliver the items to the baggage claim area. The actual time required to deliver all the items to the baggage claim area based on the number of items unloaded will be very close to the time predicted by a least-squares model. The time required to deliver an item to the baggage claim area is relatively constant, regardless of the number of baggage items unloaded from a flight. The variability in the time required to deliver all items to the baggage claim area is about the same for all flights, regardless of the number of items unloaded from a flight. The time required to unload baggage items from a flight is related to the time required to deliver the items to the baggage claim area.

The actual time required to deliver all the items to the baggage claim area based on the number of items unloaded will be very close to the time predicted by a least-squares model.

Jacques, an artisan cheese maker, collects data on every step of the cheese-making process for each batch he makes. Jacques noticed that the daily high temperature in his shop on the day he made a batch of cheese was related to the pH of the cheese the next morning. He computed the correlation between the daily high temperature and the pH of the cheese to be −0.64. What information does the correlation provide about the relationship between the daily high temperature and the pH of the cheese? The relationship is linear because the correlation is negative. The relationship is not linear because the correlation is negative. The morning pH of the cheese tends to be higher when the daily high temperature in the shop is warmer, compared to when the daily high temperature is cooler. The morning pH of the cheese tends to be higher when the daily high temperature in the shop is cooler, compared to when the daily high temperature is warmer. There is no relationship between the daily high temperature and the pH of the cheese.

The morning pH of the cheese tends to be higher when the daily high temperature in the shop is cooler, compared to when the daily high temperature is warmer.

A tennis ball was thrown in the air. The height of the ball from the ground was recorded every millisecond from the time the ball was thrown until it reached the height from which it was thrown. The correlation between the time and height was computed to be 0. What does this correlation suggest about the relationship between the time and height? There is no relationship between time and height. There is no linear relationship between time and height. The distance the ball traveled upward is the same as the distance the ball traveled downward. The correlation suggests that there is measurement or calculation error. The correlation suggests that more measurements should be taken to better understand the relationship.

There is no linear relationship between time and height.

A researcher in Alaska measured the age (in months) and the weight (in pounds) of a random sample of adolescent moose. When the least-squares regression analysis was performed, the correlation was 0.59. Which of the following is the correct way to label the correlation? 0.59 months per pound 0.59 pounds per month 0.59 0.59 months times pounds 0.59 month pounds

0.59

The least-squares regression line yˆ=1.8−0.2x summarizes the relationship between velocity, in feet per second, and depth, in feet, in measurements taken for a certain river, where x represents velocity and y represents the depth of the river. What is the predicted value of y, in feet, when x=5? −16 −1 −0.2 0.8 1.8

0.8

The height h and collar size c, both in centimeters, measured from a sample of boys were used to create the regression line cˆ=−94+0.9h. The line is used to predict collar size from height, both in centimeters, for boys' shirt collars. Which of the following has no logical interpretation in context? The predicted collar size of a boy with height 140cm The h values in the sample The c values in the sample The slope of the regression line The c-intercept of the regression line

The c-intercept of the regression line

A botanist created a linear model to predict plant height from soil acidity (pH level) for a certain type of plant. The slope of the model was 2.5 centimeters per pH level, the standard deviation of the sample of plant heights was 4 centimeters, and the standard deviation of the soil acidities was 1 pH level. What is the value of the correlation coefficient? 0.015 0.10 0.25 0.625 1.60

0.10

A restaurant manager collected data to predict monthly sales for the restaurant from monthly advertising expenses. The model created from the data showed that 36 percent of the variation in monthly sales could be explained by monthly advertising expenses. What was the value of the correlation coefficient? 0.64 0.60 0.40 0.36 0.13

0.60

A marketing consultant created a linear regression model to predict the number of units sold by a client based on the amount of money spent on marketing by the client. Which of the following is the best graphic to use to evaluate the appropriateness of the model? A dotplot A histogram A residual plot A boxplot A bar chart

A residual plot

Jordan is working on a business model for a sandwich shop. Based on past data, he developed the model nˆ=150−3p, where nˆ represents the predicted number of turkey sandwiches sold in one day for a price of p dollars per sandwich. Which of the following is the best description of the slope of the model? For each increase of $3 in the price of the sandwich, the number sold is predicted to decrease, on average, by 150. For each increase of $3 in the price of the sandwich, the number sold is predicted to increase, on average, by 150. For each increase of $1 in the price of the sandwich, the number sold is predicted to decrease, on average, by 3. For each increase of $1 in the price of the sandwich, the number sold is predicted to increase, on average, by 3. For each increase of $1 in the price of the sandwich, the number sold is predicted to decrease, on average, by 150.

For each increase of $1 in the price of the sandwich, the number sold is predicted to decrease, on average, by 3.

Dairy farmers are aware there is often a linear relationship between the age, in years, of a dairy cow and the amount of milk produced, in gallons per week. The least-squares regression line produced from a random sample is Milkˆ=40.8−1.1(Age). Based on the model, what is the difference in predicted amounts of milk produced between a cow of 5 years and a cow of 10 years? A cow of 5 years is predicted to produce 5.5 fewer gallons per week. A cow of 5 years is predicted to produce 5.5 more gallons per week. A cow of 5 years is predicted to produce 1.1 fewer gallons per week. A cow of 5 years is predicted to produce 1.1 more gallons per week. A cow of 5 years and a cow of 10 years are both predicted to produce 40.8 gallons per week.

A cow of 5 years is predicted to produce 5.5 more gallons per week.

A small business owner has created a linear regression model to predict the number of new customers who will visit a shop based on the number of times the owner has an advertisement played on the radio. What is the explanatory variable and what is the response variable? Explanatory: number of new customers; response: number of times the advertisement is played Explanatory: number of times the advertisement is played; response: number of new customers Explanatory: number of times the advertisement is played; response: number of purchases made by customers Explanatory: number of purchases made by customers; response: number of times the advertisement is played Explanatory: number of previous customers; response: number of new customers

Explanatory: number of times the advertisement is played; response: number of new customers

The least-squares regression line Sˆ=0.5+1.1L models the relationship between the listing price and the actual sales price of 12 houses, with both amounts given in hundred-thousands of dollars. Let L represent the listing price and S represent the sales price. Which of the following is the best interpretation of the slope of the regression line? For each hundred-thousand-dollar increase in the listing price, the sales price will increase by $1.1. For each hundred-thousand-dollar increase in the listing price, the sales price will increase by $110,000. For each hundred-thousand-dollar increase in the listing price, the sales price will decrease by $110,000. For each hundred-thousand-dollar increase in the listing price, the sales price is predicted to increase by $1.1. For each hundred-thousand-dollar increase in the listing price, the sales price is predicted to increase by $110,000.

For each hundred-thousand-dollar increase in the listing price, the sales price is predicted to increase by $110,000.

A marketing consultant, Sofia, has been studying the effect of increasing advertising spending on product sales. Sofia conducts several experiments, each time spending less than $1,000 in advertising. When she analyzed the relationship between x = advertising spending and y= product sales, the relationship was linear with r=0.90. Her boss is thrilled and asks her to estimate product sales for $100,000 in advertising spending. Is it appropriate for her to calculate a predicted amount of product sales with advertising spending of $100,000 ? Yes, because the association is linear. Yes, because the association is positive. Yes, because the association is strong. No, because the value of the correlation is not equal to 1. No, because $100,000 is much greater than the values used in the experiment.

No, because $100,000 is much greater than the values used in the experiment.

A family would like to build a linear regression equation to predict the amount of grain harvested per acre of land on their farm. They subdivide their land into several smaller plots of land for testing and would like to select an explanatory variable they can control. Which of the following is an appropriate explanatory variable that the family could use to create a linear regression equation? The total amount of rainfall recorded at their farm The type of crop planted in the plot the previous year The average daily temperature at their farm The variety of grain planted in the plot The amount of fertilizer applied to each plot of land

The amount of fertilizer applied to each plot of land

A restaurant manager collected data on the number of customers in a party in the restaurant and the time elapsed until the party left the restaurant. The manager computed a correlation of 0.78 between the two variables. What information does the correlation provide about the relationship between the number of customers in a party at the restaurant and the time elapsed until the party left the restaurant? The relationship is linear because the correlation is positive. The relationship is not linear because the correlation is positive. The parties with a larger number of customers are associated with the longer times elapsed until the party left the restaurant. The parties with a larger number of customers are associated with the shorter times elapsed until the party left the restaurant. There is no relationship between the number of customers in a party at a table in the restaurant and the time required until the party left the restaurant.

The parties with a larger number of customers are associated with the longer times elapsed until the party left the restaurant.

An engineer believes that there is a linear relationship between the thickness of an air filter and the amount of particulate matter that gets through the filter; that is, less pollution should get through thicker filters. The engineer tests many filters of different thickness and fits a linear model. If a linear model is appropriate, what should be apparent in the residual plot? There should be a positive, linear association in the residual plot. There should be a negative, linear association in the residual plot. All of the points must have residuals of 0. There should be no pattern in the residual plot. The residuals should have a small amount of variability for low values of the predictor variable and larger amounts of variability for high values of the predictor variable.

There should be no pattern in the residual plot.