Linear Regression and Correlation: Homework1.52008/06/17 14:04:35 GMT-52008/08/15 14:00:45.453 GMT-5BarbaraIllowskyillowskybarbara@deanza.eduSusanDeandeansusan@deanza.eduConnexionscnx@cnx.orgelementarystatisticsThis module provides a homework for Linear Regression and Correlation as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.
For each situation below, state the independent variable and the dependent variable.
aA study is done to determine if elderly drivers are involved in more motor vehicle fatalities than all other drivers. The number of fatalities per 100,000 drivers is compared to the age of drivers.bA study is done to determine if the weekly grocery bill changes based on the number of family members.cInsurance companies base life insurance premiums partially on the age of the applicant.dUtility bills vary according to power consumption.eA study is done to determine if a higher education reduces the crime rate in a population.aIndependent: Age; Dependent: FatalitiesdIndependent: Power Consumption; Dependent: Utility
In 1990 the number of driver deaths per 100,000 for the different age groups was as follows (Source: The National Highway Traffic Safety Administration's National Center for Statistics and Analysis):
AgeNumber of Driver Deaths per 100,00015-242825-391540-691070-791580+25
aFor each age group, pick the midpoint of the interval for the x value. (For the 80+ group, use 85.)bUsing “ages” as the independent variable and “Number of driver deaths per 100,000” as the dependent variable, make a scatter plot of the data.cCalculate the least squares (best–fit) line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}dFind the correlation coefficient. Is it significant?ePick two ages and find the estimated fatality rates. fUse the two points in (e) to plot the least squares line on your graph from (b).gBased on the above data, is there a linear relationship between age of a driver and driver fatality rate?
The average number of people in a family that received welfare for various years is given below. (Source: House Ways and Means Committee, Health and Human Services Department)
YearWelfare family size19694.019733.619753.219793.019833.019883.019912.9
aUsing “year” as the independent variable and “welfare family size” as the dependent variable, make a scatter plot of the data.bCalculate the least squares line. Put the equation in the form of: y^=a+bx size 12{y=a+ ital "bx"} {}cFind the correlation coefficient. Is it significant?dPick two years between 1969 and 1991 and find the estimated welfare family sizes. eUse the two points in (d) to plot the least squares line on your graph from (b).fBased on the above data, is there a linear relationship between the year and the average number of people in a welfare family?gUsing the least squares line, estimate the welfare family sizes for 1960 and 1995. Does the least squares line give an accurate estimate for those years? Explain why or why not.hAre there any outliers in the above data?iWhat is the estimated average welfare family size for 1986? Does the least squares line give an accurate estimate for that year? Explain why or why not.by^=88.7206−0.0432x size 12{y="88" "." "7206" - 0 "." "0432"x} {}c-0.8533, YesgNohNo.i2.97, YesUse the AIDS data from the practice for this section, but this time use the columns “year #” and “# new AIDS deaths in U.S.” Answer all of the questions from the practice again, using the new columns.
The height (sidewalk to roof) of notable tall buildings in America is compared to the number of stories of the building (beginning at street level). (Source: Microsoft Bookshelf)
Height (in feet)Stories1050574282836226529407906040122380381454110112710070046
aUsing “stories” as the independent variable and “height” as the dependent variable, make a scatter plot of the data.bDoes it appear from inspection that there is a relationship between the variables?cCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}dFind the correlation coefficient. Is it significant?eFind the estimated heights for 32 stories and for 94 stories.fUse the two points in (e) to plot the least squares line on your graph from (b).gBased on the above data, is there a linear relationship between the number of stories in tall buildings and the height of the buildings?hAre there any outliers in the above data? If so, which point(s)?iWhat is the estimated height of a building with 6 stories? Does the least squares line give an accurate estimate of height? Explain why or why not.jBased on the least squares line, adding an extra story adds about how many feet to a building?bYes
cy^=102.4287+11.7585x size 12{y="102" "." "4287"+"11" "." "7585"x} {}d0.9436; yese478.70 feet; 1207.73 feetgYeshYes;
57,1050 size 12{ left ("57","1050" right )} {}i172.98; Noj11.7585 feet
Below is the life expectancy for an individual born in the United States in certain years. (Source: National Center for Health Statistics)
Year of BirthLife Expectancy193059.7194062.9195070.2196569.7197371.4198274.5198775199275.7
aDecide which variable should be the independent variable and which should be the dependent variable. bDraw a scatter plot of the ordered pairs.cCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}dFind the correlation coefficient. Is it significant?eFind the estimated life expectancy for an individual born in 1950 and for one born in 1982.fWhy aren’t the answers to part (e) the values on the above chart that correspond to those years?gUse the two points in (e) to plot the least squares line on your graph from (b).hBased on the above data, is there a linear relationship between the year of birth and life expectancy?iAre there any outliers in the above data?jUsing the least squares line, find the estimated life expectancy for an individual born in 1850. Does the least squares line give an accurate estimate for that year? Explain why or why not.
The percent of female wage and salary workers who are paid hourly rates is given below for the years 1979 - 1992. (Source: Bureau of Labor Statistics, U.S. Dept. of Labor)
YearPercent of workers paid hourly rates197961.2198060.7198161.3198261.3198361.8198461.7198561.8198662.0198762.7199062.8199262.9
aUsing “year” as the independent variable and “percent” as the dependent variable, make a scatter plot of the data.bDoes it appear from inspection that there is a relationship between the variables? Why or why not?cCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}dFind the correlation coefficient. Is it significant?eFind the estimated percents for 1991 and 1988.fUse the two points in (e) to plot the least squares line on your graph from (b).gBased on the above data, is there a linear relationship between the year and the percent of female wage and salary earners who are paid hourly rates?hAre there any outliers in the above data?iWhat is the estimated percent for the year 2050? Does the least squares line give an accurate estimate for that year? Explain why or why not?bYes
cy^=−266.8863+0.1656x size 12{y= - "266" "." "8863"+0 "." "1656"x} {}d0.9448; Yese62.9206; 62.4237hNoi72.639; NoThe maximum discount value of the Entertainment® card for the “Fine Dining” section, Edition 10, for various pages is given below.
Page numberMaximum value ($)41614192515321743195715721685159017
aDecide which variable should be the independent variable and which should be the dependent variable. bDraw a scatter plot of the ordered pairs.cCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}dFind the correlation coefficient. Is it significant?eFind the estimated maximum values for the restaurants on page 10 and on page 70.fUse the two points in (e) to plot the least squares line on your graph from (b).gDoes it appear that the restaurants giving the maximum value are placed in the beginning of the “Fine Dining” section? How did you arrive at your answer?hSuppose that there were 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200? iIs the least squares line valid for page 200? Why or why not?The next two questions refer to the following data: The cost of a leading liquid laundry detergent in different sizes is given below.
Size (ounces)Cost ($)Cost per ounce163.99324.99645.9920010.99
aUsing “size” as the independent variable and “cost” as the dependent variable, make a scatter plot.bDoes it appear from inspection that there is a relationship between the variables? Why or why not?cCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}dFind the correlation coefficient. Is it significant?eIf the laundry detergent were sold in a 40 ounce size, find the estimated cost.fIf the laundry detergent were sold in a 90 ounce size, find the estimated cost.gUse the two points in (e) and (f) to plot the least squares line on your graph from (a).hDoes it appear that a line is the best way to fit the data? Why or why not?iAre there any outliers in the above data?jIs the least squares line valid for predicting what a 300 ounce size of the laundry detergent would cost? Why or why not?bYes
cy^=3.5984+0.0371x size 12{y=3 "." "5984"+0 "." "0371"x} {}d0.9986; Yese$5.08f$6.93iNojNot validaComplete the above table for the cost per ounce of the different sizes.bUsing “Size” as the independent variable and “Cost per ounce” as the dependent variable, make a scatter plot of the data.cDoes it appear from inspection that there is a relationship between the variables? Why or why not?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. Is it significant?fIf the laundry detergent were sold in a 40 ounce size, find the estimated cost per ounce.gIf the laundry detergent were sold in a 90 ounce size, find the estimated cost per ounce.hUse the two points in (f) and (g) to plot the least squares line on your graph from (b).iDoes it appear that a line is the best way to fit the data? Why or why not?jAre there any outliers in the above data?kIs the least squares line valid for predicting what a 300 ounce size of the laundry detergent would cost per ounce? Why or why not?
According to flyer by a Prudential Insurance Company representative, the costs of approximate probate fees and taxes for selected net taxable estates are as follows:
Net Taxable Estate ($)Approximate Probate Fees and Taxes ($)600,00030,000750,00092,5001,000,000203,0001,500,000438,0002,000,000688,0002,500,0001,037,0003,000,0001,350,000
aDecide which variable should be the independent variable and which should be the dependent variable.bMake a scatter plot of the data.cDoes it appear from inspection that there is a relationship between the variables? Why or why not?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. Is it significant?fFind the estimated total cost for a net taxable estate of $1,000,000. Find the cost for $2,500,000.gUse the two points in (f) to plot the least squares line on your graph from (b).hDoes it appear that a line is the best way to fit the data? Why or why not?iAre there any outliers in the above data?jBased on the above, what would be the probate fees and taxes for an estate that does not have any assets?cYesdy^=−337,424.6478+0.5463x size 12{y= - "337","424" "." "6478"+0 "." "5463"x} {}e0.9964; Yesf$208,872.49; $1,028,318.20hYesiNo
The following are advertised sale prices of color televisions at Anderson’s.
aDecide which variable should be the independent variable and which should be the dependent variable.bMake a scatter plot of the data.cDoes it appear from inspection that there is a relationship between the variables? Why or why not?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. Is it significant?fFind the estimated sale price for a 32 inch television. Find the cost for a 50 inch television.gUse the two points in (f) to plot the least squares line on your graph from (b).hDoes it appear that a line is the best way to fit the data? Why or why not?iAre there any outliers in the above data?
Below are the average heights for American boys. (Source: Physician’s Handbook, 1990)
Age (years)Height (cm)birth50.8283.8391.45106.67119.310137.114157.5
aDecide which variable should be the independent variable and which should be the dependent variable.bMake a scatter plot of the data.cDoes it appear from inspection that there is a relationship between the variables? Why or why not?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. Is it significant?fFind the estimated average height for a one year–old. Find the estimated average height for an eleven year–old.gUse the two points in (f) to plot the least squares line on your graph from (b).hDoes it appear that a line is the best way to fit the data? Why or why not?iAre there any outliers in the above data?jUse the least squares line to estimate the average height for a sixty–two year–old man. Do you think that your answer is reasonable? Why or why not?cYesdy^=65.0876+7.0948x size 12{y="65" "." "0876"+7 "." "0948"x} {}e0.9761; yesf72.2 cm; 143.13 cmhYesiNoj505.0 cm; No
The following chart gives the gold medal times for every other Summer Olympics for the women’s 100 meter freestyle (swimming).
aDecide which variable should be the independent variable and which should be the dependent variable.bMake a scatter plot of the data.cDoes it appear from inspection that there is a relationship between the variables? Why or why not?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. Is the decrease in times significant?fFind the estimated gold medal time for 1932. Find the estimated time for 1984.gWhy are the answers from (f) different from the chart values?hUse the two points in (f) to plot the least squares line on your graph from (b).iDoes it appear that a line is the best way to fit the data? Why or why not?jUse the least squares line to estimate the gold medal time for the next Summer Olympics. Do you think that your answer is reasonable? Why or why not?The next three questions use the following state information.
State# letters in nameYear entered the UnionRank for entering the UnionArea (square miles)Alabama718192252,423Colorado187638104,100Hawaii19595010,932Iowa18462956,276Maryland1788712,407Missouri18212469,709New Jersey178738,722Ohio18031744,828South Carolina131788832,008Utah18964584,904Wisconsin18483065,499
We are interested in whether or not the number of letters in a state name depends upon the year the state entered the Union.
aDecide which variable should be the independent variable and which should be the dependent variable.bMake a scatter plot of the data.cDoes it appear from inspection that there is a relationship between the variables? Why or why not?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. What does it imply about the significance of the relationship?fFind the estimated number of letters (to the nearest integer) a state would have if it entered the Union in 1900. Find the estimated number of letters a state would have if it entered the Union in 1940.gUse the two points in (f) to plot the least squares line on your graph from (b).hDoes it appear that a line is the best way to fit the data? Why or why not?iUse the least squares line to estimate the number of letters a new state that enters the Union this year would have. Can the least squares line be used to predict it? Why or why not?cNody^=47.03−0.216x size 12{y="47" "." "03" - 0 "." "216"x} {}e-0.4280f6; 5
We are interested in whether there is a relationship between the ranking of a state and the area of the state.
aLet rank be the independent variable and area be the dependent variable.bWhat do you think the scatter plot will look like? Make a scatter plot of the data.cDoes it appear from inspection that there is a relationship between the variables? Why or why not?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. What does it imply about the significance of the relationship?fFind the estimated areas for Alabama and for Colorado. Are they close to the actual areas?gUse the two points in (f) to plot the least squares line on your graph from (b).hDoes it appear that a line is the best way to fit the data? Why or why not?iAre there any outliers?jUse the least squares line to estimate the area of a new state that enters the Union. Can the least squares line be used to predict it? Why or why not?kDelete “Hawaii” and substitute “Alaska” for it. Alaska is the fortieth state with an area of 656,424 square miles. lCalculate the new least squares line.mFind the estimated area for Alabama. Is it closer to the actual area with this new least squares line or with the previous one that included Hawaii? Why do you think that’s the case?nDo you think that, in general, newer states are larger than the original states?
We are interested in whether there is a relationship between the rank of a state and the year it entered the Union.
aLet year be the independent variable and rank be the dependent variable.bWhat do you think the scatter plot will look like? Make a scatter plot of the data.cWhy must the relationship be positive between the variables?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. What does it imply about the significance of the relationship?fLet’s say a fifty-first state entered the union. Based upon the least squares line, when should that have occurred? gUsing the least squares line, how many states do we currently have? hWhy isn’t the least squares line a good estimator for this year?dy^=−480.5845+0.2748x size 12{y= - "480" "." "5845"+0 "." "2748"x} {}e0.9553f1934
Below are the percents of the U.S. labor force (excluding self-employed and unemployed ) that are members of a union. We are interested in whether the decrease is significant. (Source: Bureau of Labor Statistics, U.S. Dept. of Labor)
aLet year be the independent variable and percent be the dependent variable.bWhat do you think the scatter plot will look like? Make a scatter plot of the data.cWhy will the relationship between the variables be negative?dCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}eFind the correlation coefficient. What does it imply about the significance of the relationship?fBased on your answer to (e), do you think that the relationship can be said to be decreasing?gIf the trend continues, when will there no longer be any union members? Do you think that will happen? The next two questions refer to the following information: The data below reflects the 1991-92 Reunion Class Giving. (Source: SUNY Albany alumni magazine)
Class YearAverage GiftTotal Giving192241.67125192760.751,215193283.823,772193787.845,710194788.276,003195276.145,254195752.294,393196257.804,451197242.6818,093197649.3922,473198146.8720,997198637.0312,590
We will use the columns “class year” and “total giving” for all questions, unless otherwise stated.
aWhat do you think the scatter plot will look like? Make a scatter plot of the data.bCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}cFind the correlation coefficient. What does it imply about the significance of the relationship?dFor the class of 1930, predict the total class gift. eFor the class of 1964, predict the total class gift. fFor the class of 1850, predict the total class gift. Why doesn’t this value make any sense? by^=−569,770.2796+296.0351 size 12{y= - "569","770" "." "2796"+"296" "." "0351"} {}c0.8302d$1577.48e$11,642.68f-$22,105.33
We will use the columns “class year” and “average gift” for all questions, unless otherwise stated.
aWhat do you think the scatter plot will look like? Make a scatter plot of the data.bCalculate the least squares line. Put the equation in the form of:
y^=a+bx size 12{y=a+ ital "bx"} {}cFind the correlation coefficient. What does it imply about the significance of the relationship?dFor the class of 1930, predict the total class gift. eFor the class of 1964, predict the total class gift. fFor the class of 2010, predict the total class gift. Why doesn’t this value make any sense?Try these multiple choice questions
A correlation coefficient of -0.95 means there is a ____________ between the two variables.
AStrong positive correlationBWeak negative correlationCStrong negative correlationDNo CorrelationC
According to the data reported by the New York State Department of Health regarding West Nile Virus for the years 2000-2004, the least squares line equation for the number of reported dead birds (x size 12{x} {}) versus the number of human West Nile virus cases (y size 12{y} {}) is
y^=−10.2638+0.0491x size 12{y - ital "hat"= - "10" "." "2638"+0 "." "0491"x} {}. If the number of dead birds reported in a year is 732, how many human cases of West Nile virus can be expected?A25.7B46.2C-25.7D7513AThe next three questions refer to the following data: (showing the number of hurricanes by category to directly strike the mainland U.S. each decade) obtained from www.nhc.noaa.gov/gifs/table6.gif A major hurricane is one with a strength rating of 3, 4 or 5.
DecadeTotal Number of HurricanesNumber of Major Hurricanes 1941-195024101951-19601781961-19701461971-19801241981-19901551991-20001452001 – 200493
Using only completed decades (1941 – 2000), calculate the least squares line for the number of major hurricanes expected based upon the total number of hurricanes.
Ay^=−1.67x+0.5 size 12{y - ital "hat"= - 1 "." "67"x+0 "." 5} {}By^=0.5x−1.67 size 12{y - ital "hat"=0 "." 5x - 1 "." "67"} {}Cy^=0.94x−1.67 size 12{y - ital "hat"=0 "." "94"x - 1 "." "67"} {}Dy^=−2x+1 size 12{y - ital "hat"= - 2x+1} {}A
The correlation coefficient is 0.942. Is this considered significant? Why or why not?
ANo, because 0.942 is greater than the critical value of 0.707BYes, because 0.942 is greater than the critical value of 0.707CNo, because 0942 is greater than the critical value of 0.811DYes, because 0.942 is greater than the critical value of 0.811D
The data for 2001-2004 show 9 hurricanes have hit the mainland United States. The line of best fit predicts 2.83 major hurricanes to hit mainland U.S. Can the least squares line be used to make this prediction?
ANo, because 9 lies outside the independent variable valuesBYes, because, in fact, there have been 3 major hurricanes this decadeCNo, because 2.83 lies outside the dependent variable valuesDYes, because how else could we predict what is going to happen this decade.A