1. describe the degree of relationships betwe...
1. describe the degree of relationships between the number of absences in terms of days and grades in terms of percentage obtained by 8 individuals in statistics in research.\ngiven data:\nabsences (x): 3 5 7 9 11 13 15 17\ngrades (y): 90 88 86 84 82 80 78 76
Answer
# 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.