Question

use the given data to find the equation of the regression line. examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.
x: 10, 6, 9, 13, 7, 8, 5, 11, 14, 12, 15
y: 16.08, 8.84, 14.80, 17.78, 11.18, 13.16, 6.14, 17.00, 17.64, 17.56, 17.14
\hat{y} = 3.30 + 1.11x (round to two decimal places as needed.)
create a scatterplot of the data. choose the correct graph below.
\bigcirc a.
\bigcirc b.
\bigcirc c.
\bigcirc d.
(graphs a, b, c, d are shown with axes and data points, a has a linear trend, b has a curved trend, c has a curved trend, d has scattered points)

Explanation:

Step1: Analyze the data trend

First, we look at the given \( x \) and \( y \) values. As \( x \) increases from 5 to 15, \( y \) generally increases, but let's check the scatterplot shapes. The regression line is linear (\( \hat{y} = 3.30 + 1.11x \)), so we expect a scatterplot that has a linear trend, but also check for any non - linear patterns or clusters.

Step2: Evaluate each graph

• Graph A: Shows a linear upward trend, but let's check the data points. Wait, no, actually, when we look at the data, for \( x = 5 \), \( y = 6.14 \); \( x = 6 \), \( y = 8.84 \); \( x = 7 \), \( y = 11.18 \); \( x = 8 \), \( y = 13.16 \); \( x = 9 \), \( y = 14.80 \); \( x = 10 \), \( y = 16.08 \); \( x = 11 \), \( y = 17.00 \); \( x = 12 \), \( y = 17.56 \); \( x = 13 \), \( y = 17.78 \); \( x = 14 \), \( y = 17.64 \); \( x = 15 \), \( y = 17.14 \). So the \( y \) values increase with \( x \) up to a point (around \( x = 13 \)) and then start to decrease.
• Graph B: Shows a curved (maybe quadratic) pattern, which matches the fact that \( y \) increases then decreases.
• Graph C: Shows a curved pattern but in a different direction (increasing then curving up more? No, the data after \( x = 13 \) starts to decrease).
• Graph D: The points seem more scattered without a clear linear or the correct non - linear trend.

So the correct graph should show the increase of \( y \) with \( x \) up to a peak and then a decrease, which is graph B.

Answer:

B