1. describe the degree of relationships between the number of absences in terms of days and grades in terms…

Explanation:

Step1: Recall correlation coefficient formula

The Pearson - correlation coefficient $r$ formula is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. First, calculate the necessary sums: Let $n = 8$. $\sum x=3 + 5+7 + 9+11+13+15+17=\sum_{i = 1}^{8}x_{i}=70$. $\sum y=90 + 88+86+84+82+80+78+76=\sum_{i = 1}^{8}y_{i}=664$. $\sum xy=(3\times90)+(5\times88)+(7\times86)+(9\times84)+(11\times82)+(13\times80)+(15\times78)+(17\times76)$ $=270+440+602+756+902+1040+1170+1292 = 6472$. $\sum x^{2}=3^{2}+5^{2}+7^{2}+9^{2}+11^{2}+13^{2}+15^{2}+17^{2}=9 + 25+49+81+121+169+225+289 = 968$. $\sum y^{2}=90^{2}+88^{2}+86^{2}+84^{2}+82^{2}+80^{2}+78^{2}+76^{2}$ $=8100+7744+7396+7056+6724+6400+6084+5776 = 55284$.

Step2: Substitute values into the formula

$n(\sum xy)=8\times6472 = 51776$. $(\sum x)(\sum y)=70\times664 = 46480$. $n\sum x^{2}=8\times968 = 7744$. $(\sum x)^{2}=70^{2}=4900$. $n\sum y^{2}=8\times55284 = 442272$. $(\sum y)^{2}=664^{2}=440896$. The denominator is $\sqrt{(7744 - 4900)(442272-440896)}$ $=\sqrt{2844\times1376}=\sqrt{3913344}\approx1978.21$. The numerator is $51776 - 46480=5296$. $r=\frac{5296}{1978.21}\approx - 0.997$.

Answer:

The correlation coefficient $r\approx - 0.997$, which indicates a very strong negative linear relationship between the number of absences and the grades. That is, as the number of absences increases, the grades tend to decrease significantly.