Methods2024 Exam 1 · Q5Tech-free6 marks

Population model, transformations and confidence interval

The function $h:[0, \infty) \to \R, h(t) = \frac{3000}{t+1}$ models the population of a town after years.

Use the model $h(t)$ to predict the population of the town after four years.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

A new function, $h_1$ , models a population where $h_1(0) = h(0)$ but $h_1$ decreases at half the rate of $h$ at any point in time.

State a sequence of two transformations that maps $h$ to this new model $h_1$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.i.

In the town, 100 people were randomly selected and surveyed, with 60 people indicating that they were unhappy with the roads.

Determine an approximate 95% confidence interval for the proportion of people in the town who are unhappy with the roads. Use $z = 2$ for this confidence interval.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.ii.

A new sample of $n$ people results in the same sample proportion.

Find the smallest value of $n$ to achieve a standard deviation of $\frac{\sqrt{2}}{50}$ for the sample proportion.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Methods2024 Exam 1 · Q5Tech-free6 marks

Population model, transformations and confidence interval

The function $h:[0, \infty) \to \R, h(t) = \frac{3000}{t+1}$ models the population of a town after years.

Use the model $h(t)$ to predict the population of the town after four years.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

A new function, $h_1$ , models a population where $h_1(0) = h(0)$ but $h_1$ decreases at half the rate of $h$ at any point in time.

State a sequence of two transformations that maps $h$ to this new model $h_1$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.i.

In the town, 100 people were randomly selected and surveyed, with 60 people indicating that they were unhappy with the roads.

Determine an approximate 95% confidence interval for the proportion of people in the town who are unhappy with the roads. Use $z = 2$ for this confidence interval.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.ii.

A new sample of $n$ people results in the same sample proportion.

Find the smallest value of $n$ to achieve a standard deviation of $\frac{\sqrt{2}}{50}$ for the sample proportion.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.