3.80. Quantum tunneling. At temperatures approach- ing absolute zero ( − 273 ◦ C ) ,…

3.80. Quantum tunneling. At temperatures approach- ing absolute zero ( − 273 ◦ C ) ,…

3.80.        Quantum tunneling. At temperatures approach- ing absolute zero (−273◦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 refrig- erator 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 table.

 

TEMPERATURE,  x(◦C)                 PROPORTION OF IMPURITY, y

 

−262.0                                           .315

−265.0                                           .202

−256.0                                           .204

−267.0                                           .620

−270.0                                           .715

−272.0                                           .935

−272.4                                           .957

−272.7                                           .906

−272.8                                           .985

−272.9                                           .987

 

(a)    Find the least squares estimates of the intercept and slope. Interpret them.

(b)    Use a 95% confidence interval to estimate the slope β1. Interpret the interval in  terms  of this application. Does the interval support the hypothesis that temperature contributes infor- mation about the proportion of impurity passing through helium?

(c)    Interpret the coefficient of determination for this model.

(d)    Find a 95% prediction interval for the percent- age of impurity passing through solid helium at

−273◦C. Interpret the result.

(e)     Note that the value of x in part d is outside the

experimental region. Why might this lead to an unreliable prediction?

 

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