Least Squares Regression

1. 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

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Bankers at a large financial institution created the linear regression model above 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.

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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.

A researcher studying koi fish collected data on three variables, A, B, and C. The following residual plots show the residual for a model for predicting each variable from the age of the fish. A conclusion that a linear model between the variable and age is appropriate is supported by which plot or plots?

• The plot for variable A only

• The plot for variable B only

• The plot for variable C only

• The plots for variables A and C

• The plots for variables B and C

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

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CCSS.HSS.ID.B.6

Above is a residual plot from a regression of a variable with the independent variable x. Based on the plot, is it reasonable to conclude that a linear model is appropriate?

• Yes, because the plot shows no apparent pattern.

• Yes, because the points in the plot display less variation as

  x increases.

• Yes, because the sum of the residuals is close to zero.

• No, because the plot shows no apparent pattern.

• No, because the points in the plot display more variation as x

  x increases.

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 show above, 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

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CCSS.HSS.ID.B.6

CCSS.HSS.ID.C.7

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The least-squares regression model above 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%

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CCSS.HSS.ID.B.6

CCSS.HSS.ID.C.8

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

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CCSS.HSS.ID.B.6

CCSS.HSS.ID.C.8

In baseball, two statistics, the ERA (Earned Run Average) and the WHIP (Walks and Hits per Inning Pitched), are used to measure the quality of pitchers. For both measures, smaller values indicate higher quality. The following computer output gives the results from predicting ERA by using WHIP in a least-squares regression for the 2017 baseball season. Which of the following statements is the best interpretation of the value 6.8 shown in the output?

• ERA is predicted to increase by 6.8 units for each 1 unit increase of WHIP.

• WHIP is predicted to increase by 6.8 units for each 1 unit increase of ERA.

• For a pitcher with 0 units of WHIP, the ERA is predicted to be approximately 6.8 units.

• For a pitcher with 0 units of ERA, the WHIP is predicted to be approximately 6.8 units.

• Approximately 6.8% of the variability in ERA is due to its linear relationship with WHIP.

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