Refer to Exercise 11.14. After the least-squares line has been obtained, the table below (which is similar to Table 11.2) can be used for (1) comparing the observed and the predicted values of y and (2) computing SSE.

a. Complete the table.

b. Plot the least-squares line on a scatterplot of the data. Plot the following line on the same graph:

$\widehat{\mathbf{\text{y}}}\mathbf{\text{= 14 - 2.5x.}}$

c. Show that SSE is larger for the line in part b than for the least-squares line.

Motivation and right-oriented bias. Evolutionary theory suggests that motivated decision makers tend to exhibit a right-oriented bias. (For example, if presented with two equally valued detergent brands on a supermarket shelf, consumers are more likely to choose the brand on the right.) In Psychological Science (November 2011), researchers tested this theory using data on all penalty shots attempted in World Cup soccer matches (totaling 204 penalty shots). The researchers believed that goalkeepers, motivated to make a penalty-shot save but with little time to make a decision, would tend to dive to the right. The results of the study (percentages of dives to the left, middle, or right) are provided in the table. Note that the percentages in each row corresponding to a particular match situation add to 100%. Use graphs to illustrate the distribution of dives for the three-match situations. What inferences can you draw from the graphs?

Source: Based on M. Roskes et al., "The Right Side? Under Time Pressure, Approach Motivation Leads to Right-Oriented Bias," Psychological Science, Vol. 22, No. 11, November 2011 (adapted from Figure 2)11

Voltage sags and swells. The power quality of a transformer is measured by the quality of the voltage. Two causes of poor power quality are "sags" and "swells." A sag is an unusual dip and a swell is an unusual increase in the voltage level of a transformer. The power quality of transformers built in Turkey was investigated in Electrical Engineering (Vol. 95, 2013). For a sample of 103 transformers built for heavy industry, the mean number of sags per week was 353 and the mean number of swells per week was 184. Assume the standard deviation of the sag distribution is 30 sags per week and the standard deviation of the swell distribution is 25 swells per week.

a. For a sag distribution with any shape, what proportion of transformers will have between 263 and 443 sags per week? Which rule did you apply and why?

b. For a sag distribution that is mound-shaped and symmetric, what proportion of transformers will have between 263 and 443 sags per week? Which rule did you apply and why?

c. For a swell distribution with any shape, what proportion of transformers will have between 109 and 259 swells per week? Which rule did you apply and why?

d. For a swell distribution that is mound-shaped and symmetric, what proportion of transformers will have between 109 and 259 swells per week? Which rule did you apply and why?

The equation for a straight line (deterministic model) is

$\mathit{y}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathit{x}$

If the line passes through the point (-2, 4), then x = -2, y = 4 must satisfy the equation; that is,

$\mathbf{4}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}$

Similarly, if the line passes through the point (4, 6), then x = 4, y = 6 must satisfy the equation; that is,

$\mathbf{6}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$

Use these two equations to solve for and ; then find the equation of the line that passes through the points (-2, 4) and (4, 6).

Is honey a cough remedy? Does a teaspoon of honey before bed really calm a child’s cough? To test the folk remedy, pediatric researchers carried out a designed study conducted over two nights (Archives of Pediatrics and Adolescent Medicine, December 2007). A sample of 105 children who were ill with an upper respiratory tract infection and their parents participated in the study. On the first night, the parents rated their children’s cough symptoms on a scale from 0 (no problems at all) to 6 (extremely severe) in five different areas. The total symptoms score (ranging from 0 to 30 points) was the variable of interest for the 105 patients. On the second night, the parents were instructed to give their sick child a dosage of liquid “medicine” prior to bedtime. Unknown to the parents, some were given a dosage of dextromethorphan (DM)—an over-the-counter cough medicine—while others were given a similar dose of honey. Also, a third group of parents (the control group) gave their sick children no dosage at all. Again, the parents rated their children’s cough symptoms, and the improvement in total cough symptoms score was determined for each child. The data (improvement scores) for the study are shown in the table below, followed (in the next column) by a Minitab dot plot of the data. Notice that the green dots represent the children who received a dose of honey, the red dots represent those who got the DM dosage, and the black dots represent the children in the control group. What conclusions can pediatric researchers draw from the graph? Do you agree with the statement (extracted from the article), “Honey may be a preferable treatment for the cough and sleep difficulty associated with childhood upper respiratory tract infection”?

Congress voting on women’s issues. The American Economic Review (March 2008) published research on how the gender mix of a U.S. legislator’s children can influence the legislator’s votes in Congress. The American Association of University Women (AAUW) uses voting records of each member of Congress to compute an AAUW score, where higher scores indicate more favorable voting for women’s rights. The researcher wants to use the number of daughters a legislator has to predict the legislator’s AAUW score.

a. In this study, identify the dependent and independent variables.

b. Explain why a probabilistic model is more appropriate than a deterministic model.

c. Write the equation of the straight-line, probabilistic model.