A chemical reaction is run 12 times, and the temperature

QUESTION:

A chemical reaction is run 12 times, and the temperature \({ }^{x_{i}}\) (in \({ }^{\circ} \mathrm{C}\)) and the yield \(y_{i}\) (in percent of a theoretical maximum) is recorded each time. The following summary statistics are recorded:

\(\overline{x}=65.0\)

\(\overline{y}=29.05\)

\(\sum_{i=1}^{12}\left(x_{i}-\overline{x}\right)^{2}=6032.0\)

\(\sum_{i=1}^{12}\left(y_{i}-\overline{y}\right)^{2}=835.42\)

\(\sum_{i=1}^{12}\left(x_{i}-\overline{x}\right)\left(y_{i}-\overline{y}\right)=1998.4\)

Let \(\beta_0\) represent the hypothetical yield at a temperature of \({0}^{\circ} \mathrm{C}\), and let \(\beta_1\) represent the increase in yield caused by an increase in temperature of \({1}^{\circ} \mathrm{C}\). Assume that assumptions 1 through 4 on page 544 hold.

a. Compute the least-squares estimates \(\hat \beta_0\) and \(\hat \beta_1\).

b. Compute the error variance estimate \(s^2\).

c. Find 95% confidence intervals for \(\beta_0\) and \(\beta_1\).

d. A chemical engineer claims that the yield increases by more than 0.5 for each \({1}^{\circ} \mathrm{C}\) increase in temperature. Do the data provide sufficient evidence for you to conclude that this claim is false?

e. Find a 95% confidence interval for the mean yield at a temperature of \({40}^{\circ} \mathrm{C}\).

f. Find a 95% prediction interval for the yield of a particular reaction at a temperature of \({40}^{\circ} \mathrm{C}\).

Equation Transcription:

Text Transcription:

x_i

^oC

y_i

overline{x}=65.0

overline{y}=29.05

sum{i=1}^12(x_i-overline{x})^2=6032.0

sum{i=1}^12(y_i-overline{y})2=835.42

sum{i=1}^12(x_i-overline{x})(y_i-overline{y})=1998.4

beta_0

0^oC

beta_1

1^oC

hat{beta}_0

hat{beta}_1

s^2

beta_0

beta_1

1^oC

40^oC

40^oC