Question

Use the least squares regression line of this data set to predict a value.
Researchers at a cognitive psychology lab are studying how children develop their color vocabulary. In a recent study, the researchers showed children of various ages different colored objects and asked them to name the colors.
They recorded each child's age (in months), x, and the number of colors the child could name, y.
Age (in months) Colors
24 3
35 5
39 9
40 6
45 9
The least squares regression line of this data set is:
y=0.288x–4.145
How many colors does this line predict a 41-month-old child would be able to name?
Round your answer to the nearest integer.
colors

Answer 1

To predict the number of colors a 41-month-old child would be able to name using the least squares regression line \( y = 0.288x - 4.145 \), we simply need to substitute \( x = 41 \) into the equation.

\[
y = 0.288(41) - 4.145
\]

Calculating it step by step:

1. Calculate \( 0.288 \times 41 \):
\[
0.288 \times 41 = 11.808
\]

2. Now, substitute back into the equation:
\[
y = 11.808 - 4.145 = 7.663
\]

Finally, rounding \( 7.663 \) to the nearest integer gives:

\[
\text{Predicted colors} \approx 8
\]

Thus, the least squares regression line predicts that a 41-month-old child would be able to name approximately **8 colors**.