12.12 Homework  (Page 5/7)

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

• a Decide which variable should be the independent variable and which should be the dependent variable.
• b Make a scatter plot of the data.
• c Does it appear from inspection that there is a relationship between the variables? Why or why not?
• d Calculate the least squares line. Put the equation in the form of: $\widehat{y}=a+\text{bx}$
• e Find the correlation coefficient. Is the decrease in times significant?
• f Find the estimated gold medal time for 1932. Find the estimated time for 1984.
• g Why are the answers from (f) different from the chart values?
• h Use the two points in (f) to plot the least squares line on your graph from (b).
• i Does it appear that a line is the best way to fit the data? Why or why not?
• j 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?

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

• a Decide which variable should be the independent variable and which should be the dependent variable.
• b Make a scatter plot of the data.
• c Does it appear from inspection that there is a relationship between the variables? Why or why not?
• d Calculate the least squares line. Put the equation in the form of: $\widehat{y}=a+\text{bx}$
• e Find the correlation coefficient. What does it imply about the significance of the relationship?
• f 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.
• g Use the two points in (f) to plot the least squares line on your graph from (b).
• h Does it appear that a line is the best way to fit the data? Why or why not?
• i 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?

We are interested in whether there is a relationship between the ranking of a state and the area of the state.

• a Let rank be the independent variable and area be the dependent variable.
• b What do you think the scatter plot will look like? Make a scatter plot of the data.
• c Does it appear from inspection that there is a relationship between the variables? Why or why not?
• d Calculate the least squares line. Put the equation in the form of: $\widehat{y}=a+\text{bx}$
• e Find the correlation coefficient. What does it imply about the significance of the relationship?
• f Find the estimated areas for Alabama and for Colorado. Are they close to the actual areas?
• g Use the two points in (f) to plot the least squares line on your graph from (b).
• h Does it appear that a line is the best way to fit the data? Why or why not?
• i Are there any outliers?
• j 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?
• k Delete “Hawaii” and substitute “Alaska” for it. Alaska is the fortieth state with an area of 656,424 square miles.
• l Calculate the new least squares line.
• m 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?
• n Do you think that, in general, newer states are larger than the original states?