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The following chart gives the gold medal times for every other Summer Olympics for the women’s 100 meter freestyle (swimming).

Year Time (seconds)
1912 82.2
1924 72.4
1932 66.8
1952 66.8
1960 61.2
1968 60.0
1976 55.65
1984 55.92
1992 54.64
2000 53.8
2008 53.1

  • Decide which variable should be the independent variable and which should be the dependent variable.
  • Make a scatter plot of the data.
  • Does it appear from inspection that there is a relationship between the variables? Why or why not?
  • 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. Is the decrease in times significant?
  • Find the estimated gold medal time for 1932. Find the estimated time for 1984.
  • Why are the answers from (f) different from the chart values?
  • Use the two points in (f) to plot the least squares line on your graph from (b).
  • Does it appear that a line is the best way to fit the data? Why or why not?
  • Use 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?
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The next three questions use the following state information.

State # letters in name Year entered the Union Rank for entering the Union Area (square miles)
Alabama 7 1819 22 52,423
Colorado 1876 38 104,100
Hawaii 1959 50 10,932
Iowa 1846 29 56,276
Maryland 1788 7 12,407
Missouri 1821 24 69,709
New Jersey 1787 3 8,722
Ohio 1803 17 44,828
South Carolina 13 1788 8 32,008
Utah 1896 45 84,904
Wisconsin 1848 30 65,499

We are interested in whether or not the number of letters in a state name depends upon the year the state entered the Union.

  • Decide which variable should be the independent variable and which should be the dependent variable.
  • Make a scatter plot of the data.
  • Does it appear from inspection that there is a relationship between the variables? Why or why not?
  • 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?
  • Find 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.
  • Use the two points in (f) to plot the least squares line on your graph from (b).
  • Does it appear that a line is the best way to fit the data? Why or why not?
  • Use 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?
  • No
  • y ^ = 47 . 03 0 . 0216 x size 12{y="47" "." "03" - 0 "." "216"x} {}
  • -0.4280
  • 6; 5
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We are interested in whether there is a relationship between the ranking of a state and the area of the state.

  • Let rank be the independent variable and area be the dependent variable.
  • What do you think the scatter plot will look like? Make a scatter plot of the data.
  • Does it appear from inspection that there is a relationship between the variables? Why or why not?
  • 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?
  • Find the estimated areas for Alabama and for Colorado. Are they close to the actual areas?
  • Use the two points in (f) to plot the least squares line on your graph from (b).
  • Does it appear that a line is the best way to fit the data? Why or why not?
  • Are there any outliers?
  • Use 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?
  • Delete “Hawaii” and substitute “Alaska” for it. Alaska is the fortieth state with an area of 656,424 square miles.
  • Calculate the new least squares line.
  • Find 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?
  • Do you think that, in general, newer states are larger than the original states?
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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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