3.11 Exercises  (Page 6/6)

We are interested in whether there is a relationship between the rank of a state and the year it entered the Union.

• a Let year be the independent variable and rank be the dependent variable.
• b What do you think the scatter plot will look like? Make a scatter plot of the data.
• c Why must the relationship be positive between the variables?
• 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 Let’s say a fifty-first state entered the union. Based upon the least squares line, when should that have occurred?
• g Using the least squares line, how many states do we currently have?
• h Why isn’t the least squares line a good estimator for this year?

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 )

• a Let year be the independent variable and percent be the dependent variable.
• b What do you think the scatter plot will look like? Make a scatter plot of the data.
• c Why will the relationship between the variables be negative?
• 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 Based on your answer to (e), do you think that the relationship can be said to be decreasing?
• g If the trend continues, when will there no longer be any union members? Do you think that will happen?

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

• a What do you think the scatter plot will look like? Make a scatter plot of the data.
• b Calculate the least squares line. Put the equation in the form of: $\widehat{y}=a+\text{bx}$
• c Find the correlation coefficient. What does it imply about the significance of the relationship?
• d For the class of 1930, predict the total class gift.
• e For the class of 1964, predict the total class gift.
• f For the class of 1850, predict the total class gift. Why doesn’t this value make any sense?

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

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

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

• A Strong positive correlation
• B Weak negative correlation
• C Strong negative correlation
• D No Correlation

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 ( $x$ ) versus the number of human West Nile virus cases ( $y$ ) is $\widehat{y}=-\text{10}\text{.}\text{2638}+0\text{.}\text{0491}x$ . If the number of dead birds reported in a year is 732, how many human cases of West Nile virus can be expected?

• A 25.7
• B 46.2
• C -25.7
• D 7513

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.

• A $\widehat{y}=-1\text{.}\text{67}x+0\text{.}5$
• B $\widehat{y}=0\text{.}\mathrm{5x}-1\text{.}\text{67}$
• C $\widehat{y}=0\text{.}\text{94}x-1\text{.}\text{67}$
• D $\widehat{y}=-\mathrm{2x}+1$

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?

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