The mass of precipitate, www grams, produced in a chemical reaction and the temperature, TTT ∘C{}^{\circ}\text{C}∘C, of the catalyst used are recorded for a random sample of 6 experimental runs, as shown in the table below.
| TTT (∘C{}^{\circ}\text{C}∘C) | 155 | 165 | 175 | 185 | 195 | 205 |
|---|---|---|---|---|---|---|
| www (grams) | 85.2 | 82.1 | 78.8 | 75.9 | 72.7 | 69.5 |
State, with a reason, which variable is the explanatory variable.
A linear model for www on TTT is
w=133.85−0.3138T w = 133.85 - 0.3138T w=133.85−0.3138TGive an interpretation of the gradient of this line.
Find the value of Tˉ\bar{T}Tˉ and the value of wˉ\bar{w}wˉ.
Show that the point (Tˉ,wˉ)(\bar{T}, \bar{w})(Tˉ,wˉ) lies approximately on the given line.
Use the given line to estimate the mass of precipitate for a reaction run with a catalyst temperature of 172 ∘C{}^{\circ}\text{C}∘C.
Comment, giving a reason, on the reliability of the estimate in part (e).
The mass of precipitate in this chemical process is assumed to be normally distributed with mean 76.5 grams and standard deviation 3.8 grams.
The middle 80% of precipitate masses lies between aaa and bbb.
Find the value of aaa and the value of bbb.