Math 141 (4.1, 4.2, 4.3, and 4.4) Homework

Since outliers can greatly affect the regression line they are also called​ _______ points.

(Influential) because their presence or absence has a big effect on conclusions.

When computing the correlation​ coefficient, what is the effect of changing the order of the variables on​ r?

It has no effect on r.

The equation for the regression line relating the salary and the year first employed is given above the figure. a. Report the slope and explain what it means. b. Either interpret the​ y-intercept of​ 4,255,424 or explain why it is not appropriate to interpret the​ y-intercept.

a. Choose the correct answer below (The average salary is​ $2099 less for each year later that the person was hired or an average of​ $2099 more for each year earlier.) b. Choose the correct answer below. (The​ y-intercept of​ $4,255,424 would be the salary for a person who started in the year​ 0, which is not appropriate to interpret.)

The figure shows a scatterplot with a regression line. The data are for the 50 regions of a particular country. The predictor is the percentage of adults who​ smoke, the response is the percentage of high school students who smoke. Complete parts​ (a) and​ (b) below. Predicted Pct. Smokers=-9.605+1.618 ​(Pct. Adult​ Smoke)

a. Explain what the trend shows. Choose the correct answer below. (The higher the percentage of adults who smoke in a​ region, the higher the percentage of high school students who smoke tends to be.) b. Use the regression equation to predict the percentage of high school students who​ smoke, assuming that 20​% of adults in the region smoke. Use 20​, not 0.20. (The percentage of high school students who​ smoke, assuming that 20​% of adults in the region​ smoke, is about 23​%.)

The correlation coefficient makes sense only if the trend is linear and the​ _______.

variables are numerical.

The figure shows a scatterplot of birthrate​ (live births per 1000​ women) and age of the mother in the United States. Would it make sense to find the correlation for this data​ set? Explain. According to this​ graph, at approximately what age does the highest fertility rate​ occur? Would it make sense to find the correlation for this data​ set?

​No, it would not make sense to find the correlation because the trend is not linear.

The intercept of a regression line tells a person the predicted mean​ y-value when the​ x-value is​ _______.

0

The​ _______ is a number that measures the strength of the linear association between two numerical variables.

Correlation Coefficient

The figure shows a scatterplot of the age of students and the value of their cars. Does it show an increasing​ trend, a decreasing​ trend, a changing​ trend, or very little​ trend?

Decreasing trend

For what types of associations are regression models​ useful?

Linear

Under what conditions can extrapolation be used to make predictions beyond the range of the​ data?

Never

When can a correlation coefficient based on an observational study be used to support a claim of cause and​ effect?

Never

A large amount of scatter in a scatterplot is an indication that the association between the two variables is​ _______.

Weak (not non-linear)

The scatterplot shows data from the 50 states taken from the U.S. Censuslong dashthe percentage of the population​ (25 years or​ older) with a college degree or higher and the median family income. Describe and interpret the trend.

What is the​ trend? What does the direction of the trend​ mean? Choose the correct answer below? (The trend is positive. States with a larger proportion of​ college-educated people tend to have higher median family income.)

The figure shows the relationship between the number of miles per gallon on the highway and that in the city for some cars. a. Report the slope and explain what it means. b. Either interpret the intercept​ (7.792) or explain why it is not appropriate to interpret the intercept.

a. Select the correct choice below and fill in the answer​ box(es) to complete your choice. (For each additional city mpg, the highway value goes up by 0.9478 mpg.) b. Select the correct choice below and fill in the answer​ box(es) to complete your choice. (It is inappropriate to interpret the intercept because no cars get 0 mpg in the city.)

The figure shows a graph of the death rate in automobile accidents and the age of the driver. a. Explain what the graph tells us about drivers at different​ ages; state which ages show the safest drivers and which show the most dangerous drivers. b. Explain why it would not be appropriate to use these data for linear regression.

A. (The graph shows that young drivers and old drivers have more fatalities and that the safest drivers are between about 40 and 60 years of age.) B. (It would not be appropriate for linear regression because the trend is not linear.)

If the correlation between height and weight of a large group of people is 0.67​, find the coefficient of determination​ (as a​ percent) and explain what it means. Assume that height is the predictor and weight is the​ response, and assume that the association between height and weight is linear. Choose the correct answer below.

The coefficient of determination is 44.89​%. ​Therefore, 44.89​% of the variation in weight can be explained by the regression line. 0.67^2=0.4489 0.4489*100=44.89

The accompanying scatterplot shows a solid blue line for predicting weight from age of​ men; the dotted red line is for predicting weight from age of women. The data were collected from a large statistics class. a. Which line is higher and what does that​ mean? b. Which line has a steeper slope and what does that​ mean?

a. Choose the correct answer below. (The​ men's line is​ higher, which means that men tend to weigh more than women at all ages shown.) b. Choose the correct answer below. (The​ men's line has a steeper​ slope, which means that older men tend to outweigh younger men more than older women outweigh younger women.)

The scatterplot shows the median annual​ pay, in dollars per​ year, for​ college-educated men and women in 30 cities. The correlation is 0.885. The regression equation is shown below. Complete parts a and b. Predicted Women=12,322+0.5350 Men

a. Find a rough estimate​ (by using the​ scatterplot) of median pay for women in a city that has a median pay of about ​$50,000 for men. (Between $35,000 and $45,000) b. Use the given regression equation to get a more precise estimate of the median pay for women in a city that has a median pay for men of ​$50,000. The estimated median pay for women is ​$(39072)

The correlation coefficient is always a number between _________?

-1 and 1

​Since, in​ general, the longer a car is owned the more miles it travels one can say there is a​ _______ between age of a car and mileage.

A Positive Association

Attempting to use the regression equation to make predictions beyond the range of the data is called​ _______.

Extrapolation

The accompanying figure shows mean life expectancy versus age for males and females in a​ first-world country, up to the age of 119. Females are represented by the blue circles and males by the red squares. Complete parts​ (a) through​ (d) below.

a. Estimate the life expectancy of a 100-year old female. (3) years b. Would it make sense to find the best straight line for this​ graph? Why or why​ not? (No, linear regression would not be appropriate because the trend is not linear.) c. Is it reasonable to predict the life expectancy for a person who is 120 from the regression line for these​ data? Why or why​ not? (It is unreasonable because it would be an extrapolation.) d. Explain what it means that nearly all of the blue circles​ (for women) are above the red squares​ (for men).​ (Above the age of​ 100, the red squares cover the blue circles because both are in the same​ place.) (Women tend to live longer than men)

What type of effect can outliers have on a regression​ line?

A big effect

Refer to the scatterplot on the​ right, where the diameter of ball 1 is the independent variable and the diameter of ball 2 is the dependent variable. A. Would it make sense to find the correlation with this data​ set? Why or why​ not? B. Would the correlation be​ positive, negative, or near​ 0?

A. (No because there is no trend in the data) B. (Near zero) because there is no trend in data

Complete parts​ (a) and​ (b) below. A. The scatterplot to the right shows the college tuition and percentage acceptance at some colleges. Would it make sense to find the correlation using this data​ set? Why or why​ not? B. The scatterplot to the right shows the composite grade on the ACT​ (American College​ Testing) exam and the English grade on the same exam. Would it make sense to find the correlation using this data​ set? Why or why​ not?

A. (No. Linear regression is not appropriate because the trend is not linear.) this is because there is a curvature in the points B. (Yes. There is no reason why linear regression would not be appropriate because the trend is linear.)

A professor went to a website for rating professors and looked up the quality rating and also the​ "easiness" of the six​ full-time professors in one department. The ratings are 1​ (lowest quality) to 5​ (highest quality) and 1​ (hardest) to 5​ (easiest). The numbers given are averages for each professor. Assume the trend is​ linear, find the​ correlation, and comment on what it means. Quality Easiness 4.8 3.8 4.5 3.3 4.2 3.2 4.2 2.7 3.9 1.7 3.4 1.9

A. Calculate the correlation. r= 0.893 B. Comment on the meaning of the correlation. Choose the correct interpretation below. (The professors that have high easiness scores tend to also have high quality scores.)

Five people were asked how many female first cousins they had and how many male first cousins. The data are shown in the table. Assume the trend is​ linear, find the​ correlation, and comment on what it means. Female Male 3 5 1 1 4 3 6 9 1 1

A. R= 0.916 B. People with many female cousins tend to have many male cousins.

The scatterplot shows the number of work hours and the number of TV hours per week for some college students who work. There is a very slight trend. Is the trend positive or​ negative? What does the direction of the trend mean in this​ context? Identify any unusual points.

A. What is the​ trend? What does the direction of the trend​ mean? Choose the correct answer below. (The trend is negative. The more hours of work a student​ has, the fewer hours of TV the student tends to watch.) B. Identify any unusual points. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. (The person who works 70 70 hours appears to be an​ outlier, because that point is separated from the other points by a large amount.)

The accompanying table gives the distance from a particular city to seven other cities​ (in thousands of​ miles) and gives the time for one randomly​ chosen, commercial airplane to make that flight. Do a complete regression analysis that includes a scatterplot with the​ line, interprets the slope and​ intercept, and predicts how much time a nonstop flight from this city would take to another city that is located 3000 miles away.

Draw a scatterplot for the​ round-trip flight data. Be sure that distance is the​ x-variable and time is the​ y-variable, because time is being predicted from distance. Graph the​ best-fit line using technology. Choose the correct scatterplot below. (Scatter plot A) Does it seem that the trend is​ linear, or is there a noticeable​ curve? The linear model (is) appropriate because there is a (linear) trend in the data. Find the equation for predicting time​ (in hours) from miles​ (in thousands). Predicted Timeequals 0.78+ (1.86) Thousand Miles Interpret the slope in the context of the problem. Select the correct choice below and fill in the answer box to complete your choice. (For every additional thousand ​ miles, on​ average, the time goes up by 1.86 hours.) Interpret the intercept in the context of the problem. Although there are no flights with a distance of​ zero, try to explain what might cause the added time that the intercept represents. (A trip of zero miles would take about 0.78 hours.​ However, a trip would never be exactly zero​ miles, so this time might account for delays in taking off and landing.) Using the regression​ line, how long should it take to fly nonstop 3000​ miles? (It would​ take, on​ average, about 6.36 hours to fly 3000 miles.) 0.78+1.86*3

The​ _______ is a tool for making predictions about future observed values and is a useful way of summarizing a linear relationship.

Regression Equation

The correlation between height and arm span in a sample of adult women was found to be r equals 0.945. The correlation between arm span and height in a sample of adult men was found to be r =0.851. Which association - the association between height and arm span for​ women, or the association between height and arm span for men - is ​stronger? Explain.

The association between height and arm span for women is stronger because the value of r is farther from 0.

The figures show the number of war casualties during a certain time period and the population of some hometowns from which the servicemen or servicewomen came. Comment on the difference in graphs and in the coefficient of determination between the top scatterplot that included a major city and the bottom scatterplot that did not include that city. This city is the point with a population of nearly 4 million.

The coefficient of determination is (83.9)% including the major city and only (​4.5)% excluding the major city. Including the major city also causes the slope to (Increase significantly) This shows that the major​ city, with its large population and large number of​ casualties, (is an influential point)

When describing​ two-variable associations, a written description should always include​ trend, shape,​ strength, and which of the​ following?

The context of the data

The figure shows a scatterplot with a regression line for the average​ teacher's pay and the percentage of students graduating from high school for each state in a certain year. On the basis of the​ graph, do you think the correlation is​ positive, negative, or near​ 0? Explain what this means.

The correlation is (near zero) As the average​ teachers' pay​ increases, the percentage graduating from high school ( does not tend to change)

The scatterplot shows the actual weight and desired weight change of some students.​ Thus, if they weighed 220 and wanted to weigh​ 190, the desired weight change would be negative 30. Explain what you see. In​ particular, what does it mean that the trend is​ negative?

The more people​ weigh, the more weight they tend to want to lose.

The scatterplot shows the salary and year of first employment for some professors at a college.​ Explain, in​ context, what the negative trend shows. Who makes the most and who makes the​ least?

The people hired the most recently tend to have lower​ salaries, and the people hired many years ago tend to have higher salaries.

The scatterplot shows the median starting salaries and the median​ mid-career salaries for graduates at a selection of colleges. Complete parts​ (a) through​ (e) below. ​Mid-Career= -1797+1.792 Start Med

a. As the data are​ graphed, which is the independent and which the dependent​ variable? The independent variable is (Median Starting Salary) which is the one on bottom and the dependent variable is (Median mid-career salary) which is on the left side b. Why do you suppose median salary at a school is used instead of the​ mean? (Salary distributions are usually​ skewed, making the median a more meaningful measure of center.) c. Using the​ graph, estimate the median​ mid-career salary for a median starting salary of $ 60,000. The median​ mid-career salary for a median starting salary of $ 60,000 is about ​$(110,000) ​(Round to the nearest thousand as​ needed.) d. Use the equation to predict the median​ mid-career salary for a median starting salary of $ 60,000. The median​ mid-career salary for a median starting salary of $ 60,000 is about ​$( 110,861) e. What other factors besides starting salary might influence​ mid-career salary? Select all that apply. (The number of hours worked per week, and The amount of additional education required)

The scatterplot shows the heights of mothers and daughters. Complete parts​ (a) through​ (e) below. Daughter= 35.57+0.446 Mother

a. As the data are​ graphed, which is the independent and which the dependent​ variable? The independent variable is (Mothers height) and the dependent variable is (Daughters height) b. From the​ graph, approximate the predicted height of the daughter of a mother who is 65 inches ​(5 feetnbsp 5 inches right parenthesis tall. The predicted height of the daughter of a mother who is 65 inches tall is about (65) inches. c. From the​ equation, determine the predicted height of the daughter of a mother who is 65 inches tall. The predicted height of the daughter of a mother who is 65 inches tall is about (64.56) inches. d. Interpret the slope. Choose the correct answer below. (For each additional inch in the​ mother's height, the average​ daughter's height increases by about 0.446 inch.) e. What other factors besides​ mother's height might influence the​ daughter's height? Select all that apply. (The​ daughter's nutrition during formative years, and the fathers height)

Grades on a political science test and the number of hours of paid work in the week before the test were studied. The instructor was trying to predict the grade on a test from the hours of work. The figure shows a scatterplot and the regression line for these data. a. By looking at the plot and the line​ (without doing any​ calculations), state whether the correlation is positive or negative and explain your prediction. b. Interpret the slope. c. Interpret the intercept.

a. Choose the correct answer below. (The correlation is negative because the graph shows a decreasing trend.) b. Choose the correct answer below. (For each additional hour of​ work, the score tended to go down by 0.4817 point.) c. Choose the correct answer below. (A student who did not work would expect to get about 87 on average.

The figure shows a scatterplot with a regression line for​ teachers' average pay and the expenditure per pupil for each state for public schooling in 2007. a. From the​ graph, is the correlation between​ teachers' average pay and the expenditure per pupil positive or​ negative? b. Interpret the slope. c. Interpret the intercept or explain why it should not be interpreted.

a. Choose the correct answer below. (The correlation is positive because the graph shows an increasing trend.) b. Choose the correct answer below. (For each additional dollar spent on​ teachers' pay, the expenditure per​ pupil, on​ average, is about 23 cents higher.) c. Choose the correct answer below. (The intercept might represent the cost of education when the teachers are paid nothing.​ However, doing so requires extrapolation which may not be valid)

The data shown in the accompanying table are the number of years of formal education of the fathers and mothers of a sample of 29 statistics students at a small community college in an area with many recent immigrants. The scatterplot​ (not shown) suggests a linear trend. Complete parts a through c.

a. Find and report the regression equation for predicting the​ mother's years of education from the​ father's. Then find the predicted number of years for the mother if the father has 12 years of​ education, and find the predicted number of years for a mother if the father has 4 years of education. State the regression equation for predicting the​ mother's years of education from the​ father's. Predicted​ Mother's Education= 3.90+(0.57) Father's Education (x=variable= father, y variable= mother) Using the found regression​ line, find the predicted number of years for the mother if the father has 12 years of education. (10.74 years) 3.90+(0.57)(12)=10.74 Using the found regression​ line, find the predicted number of years for the mother if the father has 4 years of education. (6.18 years) 3.90+(0.57)(4)=6.18 b. Find and report the regression equation for predicting the​ father's years of education from the​ mother's. Then find the predicted number of years for the father if the mother has 12 years of​ education, and find the predicted number of years for the father if the mother has 4 years of education. State the regression equation for predicting the​ father's years of education from the​ mother's. Predicted​ Father's Education=3.41 +(0.57) Mothers Education (X variable= mother, Y-variable =father) Using the found regression​ line, find the predicted number of years for the father if the mother has 12 years of education. (10.25 years) (3.41+(0.57)(12)=10.25) Using the found regression​ line, find the predicted number of years for the father if the mother has 4 years of education. (5.69 years) (3.41+(0.57)(4)=5.69) c. What phenomenon does this​ demonstrate? Explain. Choose the correct answer below. (Regression toward the​ mean, because values for the predictor variable that are far from the mean lead to responses that are closer to the mean)

The accompanying table shows some data from a sample of heights of fathers and their sons. The scatterplot​ (not shown) suggests a linear trend. Complete parts​ (a) through​ (c).

a. Find and report the regression equation for predicting the​ son's height from the​ father's height. Then predict the height of a son with a father 74 inches tall. Also predict the height of a son of a father who is 65 inches tall. Write the regression equation. Predicted​ Son's Height= 20.55+(0.70) fathers height (x-variable= father, y-variable= son) Predict the height of a son with a father 74 inches tall. (72.35 inches) (20.55+0.70(74)=72.35) Predict the height of a son of a father who is 65 inches tall. (66.05 inches) (20.55+0.70(65)=66.05) b. Find and report the regression equation for predicting the​ father's height from the​ son's height. Then predict a​ father's height from that of a son who is 74 inches tall and also predict a​ father's height from that of a son who is 65 inches tall. Write the regression equation. Predicted​ Father's Height= 18.10+0.74 Son's Height (X-variable= son, y-variable=father) Predict a​ father's height from that of a son who is 74 inches tall. (72.86 inches) (18.10+0.74(74)=72.86) Predict a​ father's height from that of a son who is 65 inches tall. (66.2 inches) (19.10+0.74(65)=66.2) c. What phenomenon does this​ demonstrate? Explain. Choose the correct answer below. (Regression toward the​ mean, because values for the predictor variable that are far from the mean lead to responses that are closer to the mean.)

The accompanying table shows the number of text messages sent and received by some people in one day. Complete parts​ (a) through​ (d) below.

a. Make a scatterplot of the​ data, and state the sign of the slope from the scatterplot. Use the number sent as the independent variable. Choose the correct scatterplot below. (A) The slope of the scatterplot is (Positive) b. Use linear regression to find the equation of the​ best-fit line. Predicted (Received = 1.874+(1.010) Sent) c. Interpret the slope. Select the correct choice below and fill in the answer box to complete your choice. (For each additional message​ sent, there is an average of 1.010 more messages received.) d. Interpret the intercept and comment on it. Select the correct choice below and fill in the answer box to complete your choice. (With 0 messages​ sent, there should be about 2 ​message(s) received.)

The accompanying table shows the​ self-reported number of semesters completed and the number of units completed for 15 students at a community college. All units were​ counted, but attending summer school was not included. Complete parts​ (a) through​ (e) below.

a. Make a scatterplot with the number of semesters on the​ x-axis and the number of units on the​ y-axis. Choose the correct scatterplot below. (Scatterplot D) Does one point stand out as​ unusual? Explain why it is unusual. (The point Th ​(3​,143.0​) stands out as unusual because its​ y-value is much higher than the ​ y-values of other data points with similar​ x-values.) b. Find the numerical value for the​ correlation, including the unusual point. (The correlation is 0.696) Find the numerical value for the correlation when the unusual point is not included. (The correlation is 0.931) Comment on the difference in correlation when the unusual point is removed. (When the unusual point is removed from the data​ set, the correlation increases because the point is far from the line.) c. Report the equation of the regression​ line, including the unusual point. Predicted Units= 8.6 +(12.4) Semesters Report the equation of the regression line when the unusual point is not included. Predicted Units= -5.4 +(14.1) Semesters Comment on the difference in the equation of the regression line when the unusual point is removed. (When the unusual point is​ removed, the intercept of the equation decreases and the slope increases.) d. Insert the regression line into the scatterplot found​ earlier, including the unusual point. Use technology if possible. Choose the correct graph below. With unusual fit graph c without unusual fit graph c Comment on the difference in the regression lines when the unusual point is removed. (When the unusual point is removed from the​ data, the regression line seems to be a better fit for the points on the scatterplot.) e. Report the slope and intercept of the regression line when the unusual point is included and explain what it shows. If the intercept is not appropriate to​ report, explain why. Select the correct choice below and fill in the answer boxes to complete your choice. (The slope shows​ that, for each additional semester​ completed, the average number of units completed will increase by (12.4) The intercept shows that when the average student has completed zero semesters they will have completed (8.6) units. Report the slope and intercept of the regression line when the unusual point is not included and explain what it shows. If the intercept is not appropriate to​ report, explain why. Select the correct choice below and fill in the answer boxes to complete your choice. (The slope shows that for each additional semester​ completed, the average number of units completed will increase by (14.1) It is not appropriate to report the intercept because a student cannot complete (-5.4) units. Comment on the difference in the slope and intercept of the regression line when the unusual point is removed. (When the unusual point is​ removed, the intercept decreases and the slope increases.)

The table shows a list of the weights and prices of some turkeys at different supermarkets. Complete parts a through f.

a. Make a scatterplot with weight on the​ x-axis and cost on the​ y-axis. Choose the correct graph below. (Graph c) b. Find the numerical value for the correlation between weight and price. (r= 0.933) Explain what the positive value of the correlation means. Choose the correct answer below. (A positive correlation suggests that larger turkeys tend to have a higher price.) c. Report the equation of the best straight​ line, using weight as the predictor​ (x) and cost as the response​ (y). (predicated price= -3.644+ (1.499) weight d. Insert the line on the scatterplot. Choose the correct graph below. (Graph B) e. Report the slope and intercept of the regression line and explain what they show. If the intercept is not appropriate to​ report, explain why. Report the slope of the regression line and explain what it shows. Select the correct choice below and fill in the answer box within your choice. (for each additional pound, the price goes up by $1.5) Report the intercept of the regression line and explain what it shows. Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice. (The interpretation of the intercept is​ inappropriate, because it is not possible to have a turkey that weighs 0 pounds.) f. Add a new point to your​ data, a​ 30-pound turkey that is free. Give the new value for r and the new regression equation. Explain what the negative correlation implies. What​ happened? (State the new value for r= -0.352) Determine the new regression equation. (Predicted Price= 26.305 +(-0.516) Weight) What does the negative correlation​ imply? (A negative correlation suggests that larger turkeys tend to have a lower price.) What happened when the new data point was​ added? (The​ 30-pound free turkey was an influential​ point, which really changed the results.)

Three scatterplots are shown below. The calculated correlations are -0.03​, -0.93​, and 0.61. Determine which correlation goes with which scatterplot.

A= -0.03 B= 0.61 C= -0.93

When one has influential points in their​ data, how should regression and correlation be​ done?

(Do regression and correlation with and without these points and comment on the differences) When one has influential points in their​ data, they should do the regression and correlation with and without these points and comment on the differences.

Three scatterplots are shown below. The calculated correlations are 0.647​, -0.929​, and -0.001. Determine which correlation goes with which scatterplot.

A= -0.001 (Negative scatter but not a straight line all over the place) B= 0.647 (positive linear line) C= -0.929 (negative but has a linear line so it means that it is the most negative)

The figure shows a scatterplot of the height of the left seat of a seesaw and the height of the right seat of the same seesaw. Estimate the numerical value of the​ correlation, and explain the reason for your estimate.

The correlation is R=-1 because there is a perfect negative linear association.

The figure shows a scatterplot of shoe size and GPA for some college students. Does it show an increasing​ trend, a decreasing​ trend, or no​ trend? Is there a strong​ relationship?

The graph shows (an increasing) linear trend since the points show (an increasing) pattern by scanning the graph from left to right. The relationship (is) strong because the points (Show) consistent and small vertical spread.

Identify the predictor variable and the response variable. A teacher has data on the amount of homework given to students and the students' test scores. The teacher wants to determine her students' test scores based on the amount of homework she gave her students.

The predictor variable is the (amount of homework given) The response value is the (students' test scores)

One important use of the regression line is to do which of the​ following?

To make predictions about the values of y for a given​ x-value

Answer the questions. Complete parts​ (a) through​ (c) below.

a. What is extrapolation and why is it a bad idea in regression​ analysis? (Extrapolation is prediction far outside the range of the data. These predictions may be incorrect if the linear trend does not​ continue, and so extrapolation generally should not be trusted. ) b. How is the coefficient of determination related to the​ correlation, and what does the coefficient of determination​ show? (The coefficient of determination is the square of the​ correlation, and it shows the proportion of the variation in the response variable that is explained by the explanatory variable.) c. When testing the IQ of a group of adults​ (aged 25 to​ 50), an investigator noticed that the correlation between IQ and age was negative. Does this show that IQ goes down as we get​ older? Why or why​ not? Explain. (No, correlation does not mean causation.)

The accompanying table shows the percentage of voters in a number of cities who voted yes on a proposition to teach students primarily in​ English, and the percentage of students in each city with limited English. Assume that the association between the percentage of voters favoring the proposition and the percentage of students with limited English is linear enough to proceed. Find the regression equation for predicting​ "% Yes" from​ "% Students with Limited​ English," and report it. Interpret the sign of the slope clearly.

Determine the regression equation for predicting​ "% Yes" from​ "% Students with Limited​ English." Predicted​ % Yes=68.08+(-0.39) ​% Students with Limited English (Make sure to to x= % limited English, and y= %yes) Interpret the sign of the slope clearly. (For every additional percent of students who have limited​ English, on​ average, the percent of voters who voted yes goes down by 0.39)

The figure shows a scatterplot of the heights and weights of some women taking statistics. Describe what you see. Is the trend​ positive, negative, or near​ zero? Explain.

Does the graph show an increasing​ trend, a decreasing​ trend, or no​ trend? Choose the correct answer below. (Increasing trend) Explain what this trend means in the context of this problem. (Taller women tend to weigh more)

Answer the questions using complete sentences. a. What is an influential​ point? b. It has been noted that people who go to church frequently tend to have lower blood pressure than people who​ don't go to church. Does this mean you can lower your blood pressure by going to​ church? Why or why​ not? Explain.

a. Choose the correct answer below. (An influential point is a point that changes the regression equation by a large amount.) b. Choose the correct answer below. (Going to church may not cause lower blood pressure. Just because two variables are related does not show that one caused the other.)

The table to the right shows the number of people living in a house and the weight of trash​ (in pounds) at the curb just before trash pickup. Complete parts​ (a) through​ (c) below. People Trash (pounds) 3 30 3 22 6 90 1 30 9 94

a. Find the correlation between these numbers by using a computer or a statistical calculator. r= (0.901) b. Suppose some of the weight was from the container​ (each container weighs 5 ​pounds). Subtract 5 pounds from each​ weight, and find the new correlation with the number of people. What happens to the correlation when a constant is added​ (we added negative 5​) to each​ number? (The correlation is 0.901. The correlation coefficient remains the same when a constant is added to each number.) c. Suppose each house contained exactly twice the number of​ people, but the weight of the trash was the same. What happens to the correlation when numbers are multiplied by a​ constant? (The correlation is 0.901. The correlation coefficient remains the same when the numbers are multiplied by a positive constant.)