Question

Read the problems carefully and round answers to 3 the thousandths place where applicable.

1. A large corporation sells a sterilizer for medical tools. Engineers believe that the time to reach operational temperature (measured in minutes) is linearly related to the thickness of insulation (measured in inches). A random sample of 12 thicknesses was selected and the time to reach operational temperature was recorded. They are trying to predict time using thickness. A regression analysis was performed. The summary results are given below and the coefficient of determination was found to be 0.472. Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 8.678 0.572 15.165 0.000 7.403 9.953 Thickness -0.837 0.280 -2.990 0.014 -1.461 -0.213

a. Interpret the slope in the words of the problem.

b. Interpret the coefficient of determination in the words of the problem.

c. Find correlation coefficient and interpret its meaning in the words of the problem.

d. Is the true slope significantly different from 0? Justify!

e. If the time was 6.9 minutes for an insulation of 1.8 inches, find the residual.

Answer 1

Coefficients   Standard Error     t Stat P-value    Lower 95%      Upper 95%

Intercept         8.678              0.572             15.165     0.000       7.403              9.953

Thickness     -0.837               0.280             -2.990       0.014    -1.461            -0.213

Time = 8.678 - 0.837 * Thickness.

The coefficient of determination was found to be 0.472

a.

The slope means that if we increase the thickness by one inch, the time taken will decrease by 0.837 minutes.

b.

The coefficient of determination means 47.2% of the variation in operational temperature (measured in minutes) is explained/removed by the thickness of the insulation.

c.

Square of Determination = = 0.472.

Correlation Coefficient = r = = = 0.6870226

The two variable operational temperature and thickness of the insulation are linearly related with a strength of 0.6870226.

d.

The slope coefficient is -0.837 with p-value 0.014 is linearly significant from the zero as the p-value is less than 0.05 (alpha).

e.

The time was 6.9 minutes for insulation of 1.8 inches.

Now, the predicted time value is

Time = 8.678 - 0.837 * Thickness

= 8.678 - 0.837 * 1.8 = 7.1714

The residual is - = 6.9 - 7.1714 = -0.2714