Question: 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.

HELIUM

Proportion of Impurity

Passing Through Helium Temperature

y x (°C)

.315 -262

.202 -265

.204 -256

.620 -267

.715 - 270

.935 -272

.957 -272

.906 -272

.985 -273

.987 -273

(a) Construct a scattergram of the data.

(b) Find the least-squares line for the data and plot it on your scattergram. Define \({{\beta }_{1}}\) in the context of this problem.

(c) Test the hypothesis (at = .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.

(d) Find a 90% confidence interval for \({{\beta }_{1}}\). Interpret your results.

(e) Find the coefficient of correlation for the given data.

(f) Find the coefficient of determination for the linear model you constructed in part b. Interpret your result.

(g) Find a 99% prediction interval for the proportion of im­purity passing through helium when the temperature is set at -270°C.

(h) Estimate the mean proportion of impurity passing through helium when the temperature is set at -270°C. Use a 99% confidence interval.