Homework: 4.1/4.2 Correlation and Least-Squares Regression

An engineer wanted to determine how the weight of a car affects gas mileage. The following data represent the weight of various cars and their gas mileage. Complete parts​ (a) through​ (d). (a) Determine which variable is the likely explanatory variable and which is the likely response variable. a. The explanatory variable is the miles per gallon and the response variable is the weight. b. The explanatory variable is the weight and the response variable is the miles per gallon.

a) The explanatory variable is the weight and the response variable is the miles per gallon. b) -.880 c) Because the correlation coefficient is negative and the absolute value of the correlation​ coefficient, . 880​, is greater than the critical value for this data​ set, . 878​, a negative linear relation exists between the weight of a car and its miles per gallon

What does it mean to say that two variables are negatively​ associated? A. There is a linear relationship between the​ variables, and whenever the value of one variable​ increases, the value of the other variable increases. B. There is a linear relationship between the​ variables, and whenever the value of one variable​ increases, the value of the other variable decreases. C. There is a linear relationship between the variables. D. There is a relationship between the variables that is not linear.

B. There is a linear relationship between the​ variables, and whenever the value of one variable​ increases, the value of the other variable decreases.

What does it mean to say that two variables are positively​ associated? A. There is a relationship between the variables that is not linear. B. There is a linear relationship between the​ variables, and whenever the value of one variable​ increases, the value of the other variable increases. C. There is a linear relationship between the variables. D. There is a linear relationship between the​ variables, and whenever the value of one variable​ increases, the value of the other variable decreases.

B. There is a linear relationship between the​ variables, and whenever the value of one variable​ increases, the value of the other variable increases.

For the accompanying data​ set, (a) draw a scatter diagram of the​ data, (b) compute the correlation​ coefficient, and​ (c) determine whether there is a linear relation between x and y. x 2 6 1 7 9 y 8 7 6 9 5

C

The linear correlation between violent crime rate and percentage of the population that has a cell phone is -0.918 for years since 1995. Do you believe that increasing the percentage of the population that has a cell phone will decrease the violent crime​ rate? What might be a lurking variable between percentage of the population with a cell phone and violent crime​ rate? Will increasing the percentage of the population that has a cell phone decrease the violent crime​ rate?

No, the economy

On an international​ exam, students are asked to respond to a variety of background questions. For the 41 nations that participated in the​ exam, the correlation between the percentage of items answered in the background questionnaire​ (used as a proxy for student task​ persistence) and mean score on the exam was 0.731. Does this suggest there is a linear relation between student task persistence and achievement​ score? Write a sentence that explains what this result might mean. LOADING... Click the icon to view the critical values table. Does this suggest there is a linear relation between student task persistence and achievement​ score? Choose the best response below. A. ​No, since |0.731| is less than the critical value for 30. B. ​No, since |0.731| is greater than the critical value for 30. C. ​Yes, since |0.731| is greater than the critical value for 30. D. ​Yes, since |0.731| is less than the critical value for 30. What does this result​ mean? A. There is no linear relation between countries in which students answered a greater percentage of items in the background questionnaire and higher mean scores on the exam. B. Countries in which students answered a lesser percentage of items in the background questionnaire tended to have higher mean scores on the exam. C. Countries in which students answered a greater percentage of items in the background questionnaire tended to have higher mean scores on the exam. D. Countries in which students cheated on the exam also cheated on the items in the background questionnaire.

Yes, since |0.731| is greater than the critical value for 30. Countries in which students answered a greater percentage of items in the background questionnaire tended to have higher mean scores on the exam.

The​ least-squares regression equation is y with hat=669.8x + 16,443 where y is the median income and x is the percentage of 25 years and older with at least a​ bachelor's degree in the region. The scatter diagram indicates a linear relation between the two variables with a correlation coefficient of 0.7392. Complete parts​ (a) through​ (d).

a) ​$36537 ​ b)This is lower than expected because the expected income is ​$34327 c) For every percent increase in adults having at least a​ bachelor's degree, the median income increases by ​$669.80​, on average d) It does not make sense to interpret the​ y-intercept because an​ x-value of 0 is outside the scope of the mode

If r=​_______, then a perfect negative linear relation exists between the two quantitative variables.

-1

What is a​ residual? What does it mean when a residual is​ positive?

A residual is the difference between an observed value of the response variable y and the predicted value of y. If it is​ positive, then the observed value is greater than the predicted value. (A residual is the difference between an observed value of the response variable y and the predicted value of​ y, or the residual is the observed value minus the predicted value. That means that if the residual is​ positive, then the observed value must be greater than the predicted value.)

The data points have a linear relationship because they lie mainly in a straight line. The two variables have a positive association.

Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is​ linear, determine whether it indicates a positive or negative association between the variables. Use this information to answer the following.

The data points do not have a linear relationship because they do not lie mainly in a straight line. The relationship is not linear.

Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is​ linear, determine whether it indicates a positive or negative association between the variables. Use this information to answer the following.

Explain what each point on the​ least-squares regression line represents.

Each point on the​ least-squares regression line represents the predicted​ y-value at the corresponding value of x

True or​ false: Correlation implies causation.

False (Often times in observational​ studies, we cannot conclude two correlated variables have a causal relationship. The presence of a lurking variable that is related to both the explanatory variable and the response variable can make the two variables correlated without having a causal relation.)

(a) Scatter diagram III . ​(b) Scatter diagram I . ​(c) Scatter diagram II

Match the linear correlation coefficient to the scatter diagram. The scales on the​ x- and​ y-axis are the same for each scatter diagram. left parenthesis a right parenthesis r equals 0.787​, left parenthesis b right parenthesis r equals 1​, left parenthesis c right parenthesis r equals 0.523

If the linear correlation between two variables is​ negative, what can be said about the slope of the regression​ line? Need more information Negative Positive

Negative

Will the following variables have positive​ correlation, negative​ correlation, or no​ correlation? outside temperature and the number of people wearing coats

Negative

What does it mean if r=0?

No linear relationship exists between the variables.

Suppose that two​ variables, X and​ Y, are negatively associated. Does this mean that​ above-average values of X will always be associated with​ below-average values of​ Y? Explain

No, because association does not mean that every point fits the trend. The negative association only means that​ above-average values of X are generally associated with​ below-average values of Y.

The​ least-squares regression line always travels through the point (x-bar, y-bar) .

True

Draw a scatter diagram. Comment on the type of relation that appears to exist between x and y. ​(b) Given that x-bar=3.5000​, sx=2.2583​, y-bar 4.5000​, sy=1.6900​, and r=-0.9328​, determine the​ least-squares regression line. ​(c) Graph the​ least-squares regression line on the scatter diagram drawn in part​ (a). x 1 1 3 4 6 6 y 5.3 6.8 5.6 4.0 2.5 2.8

graph C (negative) (b) -0.698, 6.943 There appears to be a linear, negative relationship graph A (negative)