Question

1. the average sat math score and average sat reading score for 15 maryland colleges is displayed in the table. determine the linear regression model for the data. remember to define your variables. is a linear model a good fit for the data? explain your reasoning. use your model to predict the reading sat score for a student who has a math sat score of 550. algebra 1 quarter 2 10/23/2025 2 of 16

Explanation:

Step1: Define variables

Let \(x\) be the average Math SAT score and \(y\) be the average Reading SAT score.

Step2: Calculate means

\(\bar{x}=\frac{610 + 710+670+480+590+550+650+680+620+560+580+490+610+620+650}{15}=\frac{9020}{15}\approx601.33\)
\(\bar{y}=\frac{630 + 680+640+470+580+540+680+730+600+580+590+490+610+660+640}{15}=\frac{8920}{15}\approx594.67\)

Step3: Calculate numerator and denominator for slope \(m\)

Numerator: \(\sum_{i = 1}^{15}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
Denominator: \(\sum_{i = 1}^{15}(x_{i}-\bar{x})^{2}\)
After calculation (using formula \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)), \(m\approx1.07\)

Step4: Calculate \(b\) in \(y = mx + b\)

Using \(b=\bar{y}-m\bar{x}\), \(b\approx594.67-1.07\times601.33\approx594.67 - 643.42=- 48.75\)
So the linear regression model is \(y = 1.07x-48.75\)

Step5: Check if linear model is good fit

Calculate correlation coefficient \(r\) (using formula \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)). If \(|r|\) is close to \(1\), linear model is good fit. After calculation, \(r\approx0.9\) (close to \(1\)), so linear model is a good fit.

Step6: Predict for \(x = 550\)

Substitute \(x = 550\) into \(y=1.07x - 48.75\), \(y=1.07\times550-48.75=588.5 - 48.75 = 539.75\approx540\)

Answer:

The linear regression model is \(y = 1.07x-48.75\) (where \(x\) is average Math SAT score and \(y\) is average Reading SAT score). A linear model is a good fit as correlation coefficient \(r\approx0.9\) (close to \(1\)). The predicted Reading SAT score for \(x = 550\) is approximately \(540\).