10.58

Quantum tunneling . At temperatures approaching absolute zero (273 degrees below zero Celsius), helium exhibits traits that defy many laws of conventional 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 proportion (by 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 table. (file: Helium)

1. Construct a scattergram of the data.
2. Find the least-squares line for the data and plot it on your scattergram.
3. Define \[{{\beta }_{1}}\] in the context of this problem.
4. Test the hypothesis (at \[\alpha =.05\] ) that temperature contributes no information for the prediction of the proportion of impurity passing through helium when a linear model is used. Draw the appropriate conclusions.
5. Find a 90% confidence interval for \[{{\beta }_{1}}\] . Interpret your results.
6. Find the coefficient of correlation for the given data.
7. Find the coefficient of determination for the linear model you constructed in part b.
8. Find a 99% prediction interval for the proportion of impurity passing through helium when the temperature is set at -270°C.
9. Estimate the mean proportion of impurity passing through helium when the temperature is set at -270°C. Use a 99% confidence interval.

10.60

Color analysis of Saturn’s moon. High-resolution images of lapetus, one of Saturn’s largest moons, were recently obtained by the Cassini spacecraft and analyzed by NASA. Using wideband filters, the ratios of ultraviolet to green and infrared to green wavelengths were measured at 24 moon locations. These color ratios are listed in the table below. According to the researchers, "the data’s linear trend suggests mixing of two end members: Cassini Regio with a red spectrum and the south polar region with a flat spectrum." Conduct a simple linear regression analysis of the data. Do the results support the researchers’ statement? (file: Satmoon)

10.64

Snowmelt runoff erosion . The U.S. Department of Agriculture has developed and adopted the Universal Soil Loss Equation (USLE) for predicting water erosion of soils. In geographic areas where runoff from melting snow is common, calculating the USLE requires an accurate estimate of snowmelt runoff erosion. An article in the Journal of Soil and Water Conservation used simple linear regression to develop a snowmelt erosion index. Data for 54 climatologically stations in Canada were used to model the McCool winter-adjusted rainfall erosivity index, y , as a straight-line function of the once-in-5-year snowmelt runoff amount, x (measured in millimeters).

1. The data points are plotted in the scattergram shown. Is there visual evidence of a linear trend?
2. The data for seven stations were removed from the analysis due to lack of snowfall during the study period. Why is this strategy advisable?
3. The simple linear regression on the remaining n =47 data points yielded the following results: \[\hat{y}=-6.72+1.39x,{{s}_{{\hat{\beta }}}}_{_{_{1}}}=.06\] Use this information to construct a 90% confidence interval for \[{{\beta }_{1}}\] .
4. Interpret the interval, part c .

10.72

Using the 2004 cars dataset, do the following: (file: 2004 Cars)

1. Test whether the variance in City mpg is different for station wagons versus other vehicles. Use a significance level of .10
2. Use the results from the previous part to pick an appropriate test to determine whether the average City mpg for station wagons is significantly more than other vehicles. Carry out the test using a significance level of .10
3. Create a new column in the dataset containing the following revisions to the # of cylinders variable: group together 3 and 4, 5 and 6, 8 and higher – this new variable will consist of these three categories (use Calc, Recode to do this). The electric cars (-1 as the number of cylinders) will be left out of this analysis. Then test to determine whether the number of cylinders a vehicle has is dependent upon whether it is a SUV or not. Use a=.05
4. Use the dataset to investigate the linear relationship between a vehicle’s weight and its highway mpg. Also investigate the same relationship but using the reciprocal of weight instead (i.e. 1/weight). Which of these two models seems most appropriate? Be sure to explain your answer and include appropriate graphs and output.

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Deliverable: Word Document