9.140 Explain the difference between a probabilistic model and a deterministic model.

9.141 Give the general form of a straight-line model for E(y).

9.142 Outline the five steps in a simple linear regression analysis.

9.144 In fitting a least squares line to $n=15$ data points, the following quantities were computed: ${\text{SS}}_{xx}=55,$ ${\text{SS}}_{yy}=198,{\text{SS}}_{xy}=-88,\overline{x}=1.3,$ and $\bar{y}=35.$

1. Find the least squares line.

2. Graph the least squares line.

3. Calculate SSE.

4. Calculate $s^2.$

5. Find a 90% confidence interval for ${\beta }_1.$ Interpret this estimate.

6. Find a 90% confidence interval for the mean value of y when $x=15.$

7. Find a 90% prediction interval for y when $x=15.$

L09145 9.145 Consider the following sample data:

Alternate View

y	5	1	3
x	5	1	3

1. Construct a scatterplot for the data.

2. It is possible to find many lines for which $\Sigma (y-\widehat{y})=0.$ For this reason, the criterion $\Sigma (y-\widehat{y})=0$ is not used to identify the “best-fitting” straight line. Find two lines that have $\Sigma (y-\widehat{y})=0.$

3. Find the least squares line.

4. Compare the value of SSE for the least squares line with that of the two lines you found in part b. What principle of least squares is demonstrated by this comparison?

L09146 9.146 Consider the following 10 data points.

Alternate View

x	3	5	6	4	3	7	6	5	4	7
y	4	3	2	1	2	3	3	5	4	2

1. Plot the data on a scatterplot.

2. Calculate the values of r and $r^2.$

3. Is there sufficient evidence to indicate that x and y are linearly correlated? Test at the $\alpha =.10$ level of significance.

9.154 New method of estimating rainfall. Accurate measurements of rainfall are critical for many hydrological and meteorological projects. Two standard methods of monitoring rainfall use rain gauges and weather radar. Both, however, can be contaminated by human and environmental interference. In the Journal of Data Science (Apr. 2004), researchers employed artificial neural networks (i.e., computer-based mathematical models) to estimate rainfall at a meteorological station in Montreal. Rainfall estimates were made every 5 minutes over a 70-minute period by each of the three methods. The data (in millimeters) are listed in the table.

1. Propose a straight-line model relating rain gauge amount (y) to weather radar rain estimate (x).

2. Use the method of least squares to fit the model.

3. Graph the least squares line on a scatterplot of the data. Is there visual evidence of a relationship between the two variables? Is the relationship positive or negative?

4. Interpret the estimates of the y-intercept and slope in the words of the problem.

5. Find and interpret the value of s for this regression.

6. Test whether y is linearly related to x. Use $\alpha =.01$.

7. Construct a 99% confidence interval for ${\beta }_1$. Interpret the result practically.

8. Now consider a model relating rain gauge amount (y) to the artificial neural network rain estimate (x). Repeat parts a–g for this model.

SMELT 9.155 Extending the life of an aluminum smelter pot. An investigation of the properties of bricks used to line aluminum smelter pots was published in The American Ceramic Society Bulletin (Feb. 2005). Six different commercial bricks were evaluated. The life span of a smelter pot depends on the porosity of the brick lining (the less porosity, the longer is the life); consequently, the researchers measured the apparent porosity of each brick specimen, as well as the mean pore diameter of each brick. The data are given in the table.

1. a. Find the least squares line relating porosity (y) to mean pore diameter (x).

2. b. Interpret the y-intercept of the line.

3. c. Interpret the slope of the line.

4. d. Conduct a test of model adequacy. Use $\alpha =.10$.

5. e. Find r and r² and interpret these values.

6. f. Predict the apparent percentage of porosity for a brick with a mean pore diameter of 10 micrometers. Use a 90% prediction interval.

7. *g. Apply Spearman’s test for rank correlation to the data. Use $\alpha =.10.$

9.156 Relation of eye and head movements. How do eye and head movements relate to body movements when a person reacts to a visual stimulus? Scientists at the California Institute of Technology designed an experiment to answer this question and reported their results in Nature (Aug. 1998). Adult male rhesus monkeys were exposed to a visual stimulus (i.e., a panel of light-emitting diodes), and their eye, head, and body movements were electronically recorded. In one variation of the experiment, two variables were measured: active head movement (x, percent per degree) and body-­plus-head rotation (y, percent per degree). The data for $n=39$ trials were subjected to a simple linear regression analysis, with the following results: ${\widehat{\beta }}_1=.88,s_{{\widehat{\beta }}_1}=.14$

1. Conduct a test to determine whether the two variables, active head movement x and body-plus-head rotation y, are positively linearly related. Use $\alpha =.05.$

2. Construct and interpret a 90% confidence interval for ${\beta }_1.$

3. The scientists want to know whether the true slope of the line differs significantly from 1. On the basis of your answer to part b, make the appropriate inference.

CONDOR 9.157 Mortality of predatory birds. Two species of predatory birds—collard flycatchers and tits—compete for nest holes during breeding season on the island of Gotland, Sweden. Frequently, dead flycatchers are found in nest boxes occupied by tits. A field study examined whether the risk of mortality to flycatchers is related to the degree of competition between the two bird species for nest sites (The Condor, May 1995). The next table gives data on the number y of flycatchers killed at each of 14 discrete locations (plots) on the island, as well as on the nest box tit occupancy x (i.e., the percentage of nest boxes occupied by tits) at each plot. Consider the simple linear regression model $E(y)={\beta }_0+{\beta }_{1}x.$

1. Plot the data in a scatterplot. Does the frequency of flycatcher casualties per plot appear to increase linearly with increasing proportion of nest boxes occupied by tits?

2. Use the method of least squares to find the estimates of ${\beta }_0$ and ${\beta }_1$. Interpret their values.

3. Test the utility of the model, using $\alpha =.05.$

4. Find r and $r^2$ and interpret their values.

5. Find s and interpret the result.

6. Do you recommend using the model to predict the number of flycatchers killed? Explain.

9.158 Winning marathon times. In Chance (Winter 2000), statistician Howard Wainer and two students compared men’s and women’s winning times in the Boston Marathon. One of the graphs used to illustrate gender differences is reproduced below. The scatterplot graphs the winning times (in minutes) against the year in which the race was run. Men’s times are represented by purple dots and women’s times by red dots.

1. Consider only the winning times for men. Is there evidence of a linear trend? If so, propose a straight-line model for predicting winning time (y) based on year (x). Would you expect the slope of this line to be positive or negative?

2. Repeat part b for women’s times.

3. Which slope, the men’s or the women’s, will be greater in absolute value?

4. Would you recommend using the straight-line models to predict the winning time in the 2020 Boston Marathon? Why or why not?

5. Which model, the men’s or the women’s, is likely to have the smallest estimate of $\sigma$?

HELIUM 9.159 Quantum tunneling. At temperatures approaching absolute zero $(-273°\mathrm{C}),$ helium exhibits traits that seem to defy many laws of Newtonian physics. An experiment has been conducted with helium in solid form at various temperatures near absolute zero. The solid helium is placed in a dilution refrigerator along with a solid impure substance, and the fraction (in weight) of the impurity passing through the solid helium is recorded. (This phenomenon of solids passing directly through solids is known as quantum tunneling.) The data are given in the next table.

1. Find the least squares estimates of the intercept and slope. Interpret them.

2. Use a 95% confidence interval to estimate the slope ${\beta }_1.$ Interpret the interval in terms of this application. Does the interval support the hypothesis that temperature contributes information about the proportion of impurity passing through helium?

3. Interpret the coefficient of determination for this model.

4. Find a 95% prediction interval for the percentage of impurity passing through solid helium at $-273°\mathrm{C}.$ Interpret the result.

5. Note that the value of x in part d is outside the experimental region. Why might this lead to an unreliable prediction?

9.160 Dance/movement therapy. In cotherapy, two or more therapists lead a group. An article in the American Journal of Dance Therapy (Spring/Summer 1995) examined the use of cotherapy in dance/movement therapy. Two of several variables measured on each of a sample of 136 professional dance/movement therapists were years x of formal training and reported success rate y (measured as a percentage) of coleading dance/movement therapy groups.

1. Propose a linear model relating y to x.

2. The researcher hypothesized that dance/movement therapists with more years in formal dance training will report higher perceived success rates in cotherapy relationships. State the hypothesis in terms of the parameter of the model you proposed in part a.

3. The correlation coefficient for the sample data was reported as $r=-.26.$ Interpret this result.

4. Does the value of r in part c support the hypothesis in part b? Test, using $\alpha =.05.$

FLOUR 9.161 Regression through the origin. Sometimes it is known from theoretical considerations that the straight-line relationship between two variables x and y passes through the origin of the xy-plane. Consider the relationship between the total weight y of a shipment of 50-pound bags of flour and the number x of bags in the shipment. Since a shipment containing $x=0$ bags (i.e., no shipment at all) has a total weight of $y=0,$ a straight-line model of the relationship between x and y should pass through the point $x=0,y=0.$ In such a case, you could assume that ${\beta }_0=0$ and characterize the relationship between x and y with the following model:

The least squares estimate of ${\beta }_1$ for this model is

From the records of past flour shipments, 15 shipments were randomly chosen and the data shown in the following table were recorded.

1. Find the least squares line for the given data under the assumption that ${\beta }_0=0.$ Plot the least squares line on a scatterplot of the data.

2. Find the least squares line for the given data, using the model

   $$y={\beta }_0+{\beta }_{1}x+\epsilon$$

   (i.e., do not restrict ${\beta }_0$ to equal 0). Plot this line on the same scatterplot you constructed in part a.

3. Refer to part b. Why might ${\widehat{\beta }}_0$ be different from 0 even though the true value of ${\beta }_0$ is known to be 0?

4. The estimated standard error of ${\widehat{\beta }}_0$ is equal to

   $$s\sqrt{{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{\overline{x}}^2}{{\text{SS}}_x{}_x}}}$$

   Use the t-statistic

   $$t={\displaystyle \frac{{\widehat{\beta }}_0\mathrm{ –}\mathrm{ 0}}{s\sqrt{(1/n)+({\overline{x}}^2/{\text{SS}}_{xx})}}}$$

   to test the null hypothesis $H_0:{\beta }_0=0$ against the alternative $H_{\mathrm{a}}:{\beta }_0\ne 0.$ Take $\alpha =.10.$ Should you include ${\beta }_0$ in your model?

JUMP 9.162 Long-jump “takeoff error.” The long jump is a track- and-field event in which a competitor attempts to jump a maximum distance into a sandpit after a running start. At the edge of the pit is a takeoff board. Jumpers usually try to plant their toes at the front edge of this board to maximize their jumping distance. The absolute distance between the front edge of the takeoff board and the spot where the toe actually lands on the board prior to jumping is called “takeoff error.” Is takeoff error in the long jump linearly related to best jumping distance? To answer this question, kinesiology researchers videotaped the performances of 18 novice long jumpers at a high school track meet (Journal of Applied Biomechanics, May 1995). The average takeoff error x and the best jumping distance y (out of three jumps) for each jumper are recorded in the accompanying table. If a jumper can reduce his/her average takeoff error by .1 meter, how much would you estimate the jumper’s best jumping distance to change? On the basis of your answer, comment on the usefulness of the model for predicting best jumping distance.

BRICKS 9.163 Spall damage in bricks. A recent civil suit revolved around a five-building brick apartment complex located in the Bronx, New York, which began to suffer spalling damage (i.e., a separation of some portion of the face of a brick from its body). The owner of the complex alleged that the bricks were manufactured defectively. The brick manufacturer countered that poor design and shoddy management led to the damage. To settle the suit, an estimate of the rate of damage per 1,000 bricks, called the spall rate, was required (Chance, Summer 1994). The owner estimated the spall rate by using several scaffold-drop surveys. (With this method, an engineer lowers a scaffold down at selected places on building walls and counts the number of visible spalls for every 1,000 bricks in the observation area.) The brick manufacturer conducted its own survey by dividing the walls of the complex into 83 wall segments and taking a photograph of each one. (The number of spalled bricks that could be made out from each photo was recorded, and the sum over all 83 wall segments was used as an estimate of total spall damage.) In this court case, the jury was faced with the following dilemma: On the one hand, the scaffold-drop survey provided the most accurate estimate of spall rates in a given wall segment. Unfortunately, however, the drop areas were not selected at random from the entire complex; rather, drops were made at areas with high spall concentrations, leading to an overestimate of the total damage. On the other hand, the photo survey was complete in that all 83 wall segments in the complex were checked for spall damage. But the spall rate estimated by the photos, at least in areas of high spall concentration, was biased low (spalling damage cannot always be seen from a photo), leading to an underestimate of the total damage.

The data in the table are the spall rates obtained from the two methods at 11 drop locations. Use the data, as did expert statisticians who testified in the case, to help the jury estimate the true spall rate at a given wall segment. Then explain how this information, coupled with the data (not given here) on all 83 wall segments, can provide a reasonable estimate of the total spall damage (i.e., total number of damaged bricks).