3.10 Bivariate descriptive statistics: homework  (Page 12/12)

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?

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.

• a Graph a scatterplot of the data.
• b Find the correlation coefficient and determine if it is significant.
• c Find the equation of the best fit line.
• d Write the sentence that interprets the meaning of the slope of the line in the context of the data.
• e 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.)
• f Accurately graph the best fit line on your scatterplot.
• g 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?
• h Identify any outliers, using either the graphical or numerical procedure demonstrated in the textbook.
• i 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.
• j 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.

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.

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:

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

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 (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 ( $x$ ) 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? $r=0.5490$

• A No prediction can be made.
• B 19.6
• C 15
• D 38.1

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.