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
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.
Which scatter plot shows the trend in sales over time from when the app was first released?
Loading Graph...ALoading Graph...BLoading Graph...CThe function $y=1000\left(50^{\frac{t}{10}}-1\right)$y=1000(50t10−1) 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(50t10−1).
$t$t $0$0 $10$10 $y$y $\editable{}$ $\editable{}$ $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(50110t−1), what were CTech’s sales one month after BTech released their new app?