Question

A chemical reaction time y (hr) is related to the temperature (°F) in the reaction vessel. The reaction takes place according to the simple linear regression equation: y = 4.00 - .01x and ? = 0.080. The detailed explanations are needed in all parts.

a. What is the probability that the time to failure exceed 1.8 hr when the applied temperature is 230°F.

b. What is the expected change in reaction time for a 1°F increase in temperature? Explain in detailed how the equation is simplified.

c. Suppose five observations are made independently on reaction time, each one for a temperature of 250°F. What is the probability that all five times are between 1.3 and 1.7 hr?

d. What is the probability that two independently observed reaction times for temperatures 1°F apart are such that the time at the higher temperature exceeds the time at the lower temperature?

Answer 1

a)

expected reaction time when temp is 230 oF=230*-0.01+4=

1.7

  probability that the time to failure exceed 1.8 hr when the applied temperature is 230°F:

P(X>1.8)=P(Z>(1.8-1.7)/0.08)=P(Z>1.25)=0.1056

b)

expected change in reaction time for a 1°F increase in temperature =1*slope=1*(-0.01) = -0.01

c)

expected reaction time when temp is 250 oF=250*-0.01+4=

1.5

P(1.3<X<1.7)=P((1.3-1.5)/0.08<Z<(1.7-1.5)/0.08)=P(-2.5<Z<2.5)=			0.9876
hence all 5 observations are between 1.3 and 1.7=0.9876^5=			0.9395

d)

here estimated difference in flow rate=		-0.0100
and standard deviation of estimated difference=√((0.08^2+0.08^2)=0.1131
P(Y>0)=P(Z>(0-0.01)/0.1131)=P(Z>0.09)=			0.4641