Question

number of new cases for 2004, rounded to the nearest whole nu
years since 1996 (x) new cases (y)
0 757
1 719
2 682
3 705
4 645
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answer attempt 1 out of 2
regression equation:
final answer:
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Explanation:

Step1: Calculate necessary sums

First, we need to calculate the sums of \( x \), \( y \), \( xy \), and \( x^2 \).
Given \( x = [0, 1, 2, 3, 4] \) and \( y = [757, 719, 682, 705, 645] \)

• \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
• \( \sum y = 757 + 719 + 682 + 705 + 645 = 3508 \)
• \( \sum xy = (0\times757)+(1\times719)+(2\times682)+(3\times705)+(4\times645)= 0 + 719 + 1364 + 2115 + 2580 = 6778 \)
• \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)

Step2: Calculate slope (\( m \)) and intercept (\( b \)) for linear regression

The formula for the slope \( m \) of the linear regression line \( y = mx + b \) is:
\( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2} \)
where \( n = 5 \) (number of data points)
Substitute the values:
\( m=\frac{5\times6778 - 10\times3508}{5\times30 - 10^2}=\frac{33890 - 35080}{150 - 100}=\frac{-1190}{50}=-23.8 \)

The formula for the intercept \( b \) is:
\( b=\frac{\sum y - m\sum x}{n} \)
Substitute the values:
\( b=\frac{3508 - (-23.8)\times10}{5}=\frac{3508 + 238}{5}=\frac{3746}{5}=749.2 \)

Step3: Determine the year for 2004

Since \( x \) is years since 1996, for 2004, \( x = 2004 - 1996 = 8 \)

Step4: Predict the number of new cases

Using the regression equation \( y = -23.8x + 749.2 \), substitute \( x = 8 \)
\( y=-23.8\times8 + 749.2=-190.4 + 749.2 = 558.8\approx559 \)

Answer:

The regression equation is \( y = -23.8x + 749.2 \) and the number of new cases in 2004 is approximately \( \boxed{559} \)