Question

unit 3: two variable data
l4 practice – lsrl
name:

1. high school statistics - fall semester mid-term grades

absences | 17 | 0 | 5 | 10 | 18 | 5 | 0 | 0 | 2 | 3 | 6 | 9 | 19 | 18 | 0 | 2 | 3 | 18 | 6
class grade | 73 | 90 | 90 | 92 | 68 | 89 | 94 | 97 | 86 | 100 | 92 | 68 | 71 | 65 | 94 | 89 | 84 | 76 | 63
define each variable for this investigation. what data will x & y represent?
explanatory variable:
response variable:
predict the strength and trend of this relationship, providing reasoning in context of the two variables.
strong positive - or - strong negative - or - weak positive - or weak negative
explanation:
round to thousandths for any calculations below!
equation of lsrl | ŷ =
slope of lsrl *identify the value and interpret |
y-intercept of lsrl *identify the value and interpret |
coefficient of determination *identify the value and interpret |
correlation coefficient *identify the value and interpret |

Explanation:

Step1: Organize the data

Let \( x \) be the number of absences and \( y \) be the class grade. The data points are:
\( (17, 73), (0, 90), (5, 90), (10, 92), (18, 68), (5, 89), (0, 94), (0, 97), (2, 86), (3, 100), (6, 92), (9, 68), (19, 71), (18, 65), (0, 94), (2, 89), (3, 84), (18, 76), (6, 63) \)

Step2: Calculate necessary sums

First, calculate \( n = 19 \) (number of data points).
\( \sum x = 17 + 0 + 5 + 10 + 18 + 5 + 0 + 0 + 2 + 3 + 6 + 9 + 19 + 18 + 0 + 2 + 3 + 18 + 6 = 129 \)
\( \sum y = 73 + 90 + 90 + 92 + 68 + 89 + 94 + 97 + 86 + 100 + 92 + 68 + 71 + 65 + 94 + 89 + 84 + 76 + 63 = 1634 \)
\( \sum xy = 17\times73 + 0\times90 + 5\times90 + 10\times92 + 18\times68 + 5\times89 + 0\times94 + 0\times97 + 2\times86 + 3\times100 + 6\times92 + 9\times68 + 19\times71 + 18\times65 + 0\times94 + 2\times89 + 3\times84 + 18\times76 + 6\times63 \)
\( = 1241 + 0 + 450 + 920 + 1224 + 445 + 0 + 0 + 172 + 300 + 552 + 612 + 1349 + 1170 + 0 + 178 + 252 + 1368 + 378 = 10613 \)
\( \sum x^2 = 17^2 + 0^2 + 5^2 + 10^2 + 18^2 + 5^2 + 0^2 + 0^2 + 2^2 + 3^2 + 6^2 + 9^2 + 19^2 + 18^2 + 0^2 + 2^2 + 3^2 + 18^2 + 6^2 \)
\( = 289 + 0 + 25 + 100 + 324 + 25 + 0 + 0 + 4 + 9 + 36 + 81 + 361 + 324 + 0 + 4 + 9 + 324 + 36 = 1957 \)

Step3: Calculate the slope (\( m \)) of LSRL

The formula for the slope of the least - squares regression line is \( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} \)
Substitute the values:
\( n = 19 \), \( \sum xy = 10613 \), \( \sum x = 129 \), \( \sum y = 1634 \), \( \sum x^{2}=1957 \)
\( m=\frac{19\times10613 - 129\times1634}{19\times1957-(129)^{2}}=\frac{201647-209786}{37183 - 16641}=\frac{- 8139}{20542}\approx - 0.396 \)

Step4: Calculate the y - intercept (\( b \)) of LSRL

The formula for the y - intercept is \( b=\frac{\sum y - m\sum x}{n} \)
\( b=\frac{1634-(-0.396)\times129}{19}=\frac{1634 + 51.084}{19}=\frac{1685.084}{19}\approx88.689 \)

Step5: Equation of LSRL

The equation of the least - squares regression line is \( \hat{y}=mx + b \), so \( \hat{y}=-0.396x + 88.689 \)

Step6: Coefficient of Determination (\( r^{2} \)) and Correlation Coefficient (\( r \))

First, calculate the correlation coefficient \( r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}} \)
We need to calculate \( \sum y^{2}=73^{2}+90^{2}+90^{2}+92^{2}+68^{2}+89^{2}+94^{2}+97^{2}+86^{2}+100^{2}+92^{2}+68^{2}+71^{2}+65^{2}+94^{2}+89^{2}+84^{2}+76^{2}+63^{2} \)
\( = 5329+8100 + 8100+8464+4624+7921+8836+9409+7396+10000+8464+4624+5041+4225+8836+7921+7056+5776+3969 = 143330 \)
\( n\sum y^{2}-(\sum y)^{2}=19\times143330-(1634)^{2}=2723270 - 2669956 = 53314 \)
\( r=\frac{19\times10613-129\times1634}{\sqrt{(19\times1957 - 129^{2})(19\times143330 - 1634^{2})}}=\frac{-8139}{\sqrt{20542\times53314}}\approx\frac{-8139}{\sqrt{1095200000}}\approx\frac{-8139}{33094}\approx - 0.246 \)
The coefficient of determination \( r^{2}=(-0.246)^{2}\approx0.060 \)

Answer:

• Explanatory Variable: \( x \) represents the number of absences (quantitative variable, used to explain changes in class grade)
• Response Variable: \( y \) represents the class grade (quantitative variable, the variable we are trying to predict based on the number of absences)
• Trend and Strength: Weak Negative. As the number of absences (\( x \)) increases, the class grade (\( y \)) tends to decrease, but the relationship is not very strong (since \( |r|\approx0.246<0.5 \))
• Equation of LSRL: \( \hat{y}=-0.396x + 88.689 \)
• Slope of LSRL: \( m=-0.396 \). Interpretation: For each additional absence, the predicted class grade decreases by approximately 0.396 points.
• Y - Intercept of LSRL: \( b = 88.689 \). Interpretation: When the number of absences is 0, the predicted class grade is approximately 88.689 points.
• Coefficient of Determination: \( r^{2}\approx0.060 \). Interpretation: Approximately 6.0% of the variation in class grades can be explained by the linear relationship with the number of absences.
• Correlation Coefficient: \( r\approx - 0.246 \). Interpretation: There is a weak negative linear relationship between the number of absences and class grades.