10

10. [Maximum mark: 15]

The following table shows the population of Canada $t$ years after the year 2000.

[Table provided with t values 0, 5, 10, 15, 20 and corresponding p values 30.7, 32.2, 34.0, 35.7, 37.9]

A student uses linear regression to model the population of Canada using these data. The student model is $p = at + b$.

(a) (i) Write down the value of $a$ and the value of $b$.

(ii) Interpret, in context, the value of $a$. [3]

The student uses this model to predict the population of Canada in the year 2030, where $t = 30$, and calculates a population of approximately 41.3 million people.

(b) Comment on the reliability of the student’s prediction. [1]

A data scientist, Benoit, uses additional information to develop an exponential model for Canada’s future population.

In this model, $B(t) = 33.5(1.005)^t$ represents the millions of people in Canada $t$ years after the year 2000, where $25 \le t \le 100$.

(c) (i) Use Benoit’s model to predict the population of Canada in the year 2100.

(ii) Interpret, in context, the value 1.005 in Benoit’s model. [3]

Another data scientist, Cecilia, develops a third model for the Canadian population.

In this model, $C(t) = \frac{62}{1 + e^{-0.02t}}$ represents the millions of people in Canada $t$ years after the year 2000, where $25 \le t \le 100$.

(d) Use Cecilia’s model to predict the population of Canada in the year 2100. [1]

(e) Determine the year in which the difference between the predictions from Benoit’s model and Cecilia’s model is greatest. [3]

(f) Find the value of
(i) $B'(75)$;

(ii) $C'(75)$. [2]

(g) Compare and interpret, in context, the values of $B'(75)$ and $C'(75)$. [2]