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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 )

Year Percent
1945 35.5
1950 31.5
1960 31.4
1970 27.3
1980 21.9
1993 15.8
2011 11.8

  • Let year be the independent variable and percent be the dependent variable.
  • What do you think the scatter plot will look like? Make a scatter plot of the data.
  • Why will the relationship between the variables be negative?
  • Calculate the least squares line. Put the equation in the form of: y ^ = a + bx size 12{y=a+ ital "bx"} {}
  • Find the correlation coefficient. What does it imply about the significance of the relationship?
  • Based on your answer to (e), do you think that the relationship can be said to be decreasing?
  • If 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 Year Average Gift Total Giving
1922 41.67 125
1927 60.75 1,215
1932 83.82 3,772
1937 87.84 5,710
1947 88.27 6,003
1952 76.14 5,254
1957 52.29 4,393
1962 57.80 4,451
1972 42.68 18,093
1976 49.39 22,473
1981 46.87 20,997
1986 37.03 12,590

We will use the columns “class year” and “total giving” for all questions, unless otherwise stated.

  • What do you think the scatter plot will look like? Make a scatter plot of the data.
  • Calculate the least squares line. Put the equation in the form of: y ^ = a + bx size 12{y=a+ ital "bx"} {}
  • Find the correlation coefficient. What does it imply about the significance of the relationship?
  • For the class of 1930, predict the total class gift.
  • For the class of 1964, predict the total class gift.
  • For the class of 1850, predict the total class gift. Why doesn’t this value make any sense?
  • y ^ = 569 , 770 . 2796 + 296 . 0351 x size 12{y= - "569","770" "." "2796"+"296" "." "0351"} {}
  • 0.8302
  • $1577.46
  • $11,642.66
  • -$22,105.34

We will use the columns “class year” and “average gift” for all questions, unless otherwise stated.

  • What do you think the scatter plot will look like? Make a scatter plot of the data.
  • Calculate the least squares line. Put the equation in the form of: y ^ = a + bx size 12{y=a+ ital "bx"} {}
  • Find the correlation coefficient. What does it imply about the significance of the relationship?
  • For the class of 1930, predict the average class gift.
  • For the class of 1964, predict the average class gift.
  • For the class of 2010, predict the average class gift. Why doesn’t this value make any sense?

We are interested in exploring the relationship between the weight of a vehicle and its fuel efficiency (gasoline mileage). The data in the table show the weights, in pounds, and fuel efficiency, measured in miles per gallon, for a sample of 12 vehicles.

Weight Fuel Efficiency
2715 24
2570 28
2610 29
2750 38
3000 25
3410 22
3640 20
3700 26
3880 21
3900 18
4060 18
4710 15

  • Graph a scatterplot of the data.
  • Find the correlation coefficient and determine if it is significant.
  • Find the equation of the best fit line.
  • Write the sentence that interprets the meaning of the slope of the line in the context of the data.
  • What percent of the variation in fuel efficiency is explained by the variation in the weight of the vehicles, using the regression line? (State your answer in a complete sentence in the context of the data.)
  • Accurately graph the best fit line on your scatterplot.
  • For the vehicle that weights 3000 pounds, find the residual (y-yhat). Does the value predicted by the line underestimate or overestimate the observed data value?
  • Identify any outliers, using either the graphical or numerical procedure demonstrated in the textbook.
  • The outlier is a hybrid car that runs on gasoline and electric technology, but all other vehicles in the sample have engines that use gasoline only. Explain why it would be appropriate to remove the outlier from the data in this situation. Remove the outlier from the sample data. Find the new correlation coefficient, coefficient of determination, and best fit line.
  • Compare the correlation coefficients and coefficients of determination before and after removing the outlier, and explain in complete sentences what these numbers indicate about how the model has changed.

  • r = -0.8, significant
  • yhat = 48.4-0.00725x
  • For every one pound increase in weight, the fuel efficiency tends to decrease (or is predicted to decrease) by 0.00725 miles per gallon. (For every one thousand pounds increase in weight, the fuel efficiency tends to decrease by 7.25 miles per gallon.)
  • 64% of the variation in fuel efficiency is explained by the variation in weight using the regression line.
  • yhat=48.4-0.00725(3000)=26.65 mpg. y-yhat=25-26.65=-1.65. Because yhat=26.5 is greater than y=25, the line overestimates the observed fuel efficiency.
  • (2750,38) is the outlier. Be sure you know how to justify it using the requested graphical or numerical methods, not just by guessing.
  • yhat = 42.4-0.00578x
  • Without outlier, r=-0.885, rsquare=0.76; with outlier, r=-0.8, rsquare=0.64. The new linear model is a better fit, after the outlier is removed from the data, because the new correlation coefficient is farther from 0 and the new coefficient of determination is larger.

The four data sets below were created by statistician Francis Anscomb. They show why it is important to examine the scatterplots for your data, in addition to finding the correlation coefficient, in order to evaluate the appropriateness of fitting a linear model.

Set 1 Set 2 Set 3 Set 4
x y x y x y x y
10 8.04 10 9.14 10 7.46 8 6.58
8 6.95 8 8.14 8 6.77 8 5.76
13 7.58 13 8.74 13 12.74 8 7.71
9 8.81 9 8.77 9 7.11 8 8.84
11 8.33 11 9.26 11 7.81 8 8.47
14 9.96 14 8.10 14 8.84 8 7.04
6 7.24 6 6.13 6 6.08 8 5.25
4 4.26 4 3.10 4 5.39 19 12.50
12 10.84 12 9.13 12 8.15 8 5.56
7 4.82 7 7.26 7 6.42 8 7.91
5 5.68 5 4.74 5 5.73 8 6.89

a. For each data set, find the least squares regression line and the correlation coefficient. What did you discover about the lines and values of r?

For each data set, create a scatter plot and graph the least squares regression line. Use the graphs to answer the following questions:

  • For which data set does it appear that a curve would be a more appropriate model than a line?
  • Which data set has an influential point (point close to or on the line that greatly influences the best fit line)?
  • Which data set has an outlier (obviously visible on the scatter plot with best fit line graphed)?
  • Which data set appears to be the most appropriate to model using the least squares regression line?

a. All four data sets have the same correlation coefficient r=0.816 and the same least squares regression line yhat=3+0.5x

b. Set 2 ; c. Set 4 ; d. Set 3 ; e. Set 1

Try these multiple choice questions

A correlation coefficient of -0.95 means there is a ____________ between the two variables.

  • Strong positive correlation
  • Weak negative correlation
  • Strong negative correlation
  • No Correlation

C

According to the data reported by the New York State Department of Health regarding West Nile Virus (http://www.health.state.ny.us/nysdoh/westnile/update/update.htm) for the years 2000-2008, 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 . 0491 x size 12{y - ital "hat"= "19" "." "2399"+0 "." "0257"x} {} . If the number of dead birds reported in a year is 732, how many human cases of West Nile virus can be expected? r = 0.5490

  • No prediction can be made.
  • 19.6
  • 15
  • 38.1

A

The 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.

Decade Total Number of Hurricanes Number of Major Hurricanes
1941-1950 24 10
1951-1960 17 8
1961-1970 14 6
1971-1980 12 4
1981-1990 15 5
1991-2000 14 5
2001 – 2004 9 3

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.

  • y ^ = 1 . 67 x + 0 . 5 size 12{y - ital "hat"= - 1 "." "67"x+0 "." 5} {}
  • y ^ = 0 . 5x 1 . 67 size 12{y - ital "hat"=0 "." 5x - 1 "." "67"} {}
  • y ^ = 0 . 94 x 1 . 67 size 12{y - ital "hat"=0 "." "94"x - 1 "." "67"} {}
  • y ^ = 2x + 1 size 12{y - ital "hat"= - 2x+1} {}

B

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?

  • No, because 9 lies outside the independent variable values
  • Yes, because, in fact, there have been 3 major hurricanes this decade
  • No, because 2.83 lies outside the dependent variable values
  • Yes, because how else could we predict what is going to happen this decade.

A

**Exercises 42 and 43 contributed by Roberta Bloom

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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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