 New Zealand
Level 7 - NCEA Level 2

Fitting functions to Data

Lesson

As we saw, bivariate data isn't always linear and sometimes we need to consider fitting something other than a linear model to our data.

In our chapter on data transformations we saw we could do this by adjusting our data to make it more linear.

Since we have excellent technology at our disposal, we can leave the data as is and instead fit a different regression model to our data if we need to.

Using technology to fit other regression models

Let's consider the following data set from the previous chapter.

$x$x $4$4 $4.8$4.8 $5.1$5.1 $6$6 $7.1$7.1 $8.2$8.2 $9.4$9.4
$y$y $19.4$19.4 $20.4$20.4 $20.2$20.2 $19.1$19.1 $18$18 $14.9$14.9 $10$10

As we can see from the scattergraph, the data looks parabolic. Instead of transforming the data, let's have our calculator fit a quadratic regression model to the data.

As you can see, when I go to choose a model for regression, there are many to choose from. Your knowledge of functions will help you make the best choice. And here we have the equation of the quadratic function fitted to the data. We can see the value of the coefficient of determination, $r^2$r2, is very strong, and we have the quadratic function $y=-0.52x^2+5.27x+6.86$y=0.52x2+5.27x+6.86

Worked Examples

Question 1

Iain made some syrup and then set it aside to cool down. He measured the temperature of the syrup at $2$2-minute intervals for $10$10 minutes. The temperature each time is represented in the scatter plot.

1. The relationship that models the temperature of the syrup over time is best modelled by:

A linear function

A

An exponential function

B

A linear function

A

An exponential function

B
2. The function $y=75+105\times10^{-0.08t}$y=75+105×100.08t is used to model the relationship and has been graphed on the same set of axes as the plotted points.

Why is the function suitable to model the relationship?

Most of the points corresponding to actual measurements lie on the graph of the function.

A

Only an exponential function could be used to model the behaviour that as time increases, the temperature decreases.

B

Most of the points corresponding to actual measurements lie on the graph of the function.

A

Only an exponential function could be used to model the behaviour that as time increases, the temperature decreases.

B
3. After $20$20 minutes, the syrup has cooled enough to eat. At what temperature is the syrup cool enough to eat? Give your answer correct to one decimal place.

Question 2

When CTech first released a digital application (an ‘app’) onto the market, the number of sales increased slowly at first, but then the number of sales started to increase very rapidly.

1. Which scatter plot shows the trend in sales over time from when the app was first released?

A

B

C

A

B

C
2. The function $y=1000\left(50^{\frac{t}{10}}-1\right)$y=1000(50t101) is used to approximate the number of sales after $t$t months, where $y$y represents the number of sales.

Complete the table of values for $y=1000\left(50^{\frac{t}{10}}-1\right)$y=1000(50t101).

$t$t $y$y $0$0 $10$10 $\editable{}$ $\editable{}$
3. $20$20 months after CTech released their app, a rival company, BTech, released a similar app with improved features. In the month that followed, CTech’s sales dropped by $60000$60000 from their previous month's sales. According to the model $y=1000\left(50^{\frac{1}{10}t}-1\right)$y=1000(50110t1), what were CTech’s sales one month after BTech released their new app?

Outcomes

S7-2

S7-2 Make inferences from surveys and experiments: A making informal predictions, interpolations, and extrapolations B using sample statistics to make point estimates of population parameters C recognising the effect of sample size on the variability of an estimate

91264

Use statistical methods to make an inference