Question

here is a bivariate data set. find the regression equation for the response variable y.

x	y
42.6	7.5
56.5	-29.2
29.2	94.5
46.1	9.6
38	35.7
10	110.1
70	54.1
35	99

regression equation:

enter the equation in slope-intercept form with parameters accurate to two decimal places. (remember to include \y = \

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Explanation:

Step1: Calculate necessary sums

First, we need to calculate the sums of \( x \), \( y \), \( xy \), and \( x^2 \) for the given data points. Let's list the data points:
\( (x_1, y_1)=(48.4, 38) \), \( (x_2, y_2)=(42.6, 7.5) \), \( (x_3, y_3)=(56.5, -29.2) \), \( (x_4, y_4)=(29.2, 94.5) \), \( (x_5, y_5)=(46.1, 9.6) \), \( (x_6, y_6)=(38, 35.7) \), \( (x_7, y_7)=(10, 110.1) \), \( (x_8, y_8)=(70, 54.1) \), \( (x_9, y_9)=(35, 99) \)

Number of data points \( n = 9 \)

Calculate \( \sum x \):
\( \sum x=48.4 + 42.6+56.5 + 29.2+46.1+38+10+70+35 \)
\( = 48.4+42.6 = 91; 91+56.5 = 147.5; 147.5+29.2 = 176.7; 176.7+46.1 = 222.8; 222.8+38 = 260.8; 260.8+10 = 270.8; 270.8+70 = 340.8; 340.8+35 = 375.8 \)

Calculate \( \sum y \):
\( \sum y=38 + 7.5-29.2+94.5+9.6+35.7+110.1+54.1+99 \)
\( 38+7.5 = 45.5; 45.5 - 29.2=16.3; 16.3+94.5 = 110.8; 110.8+9.6 = 120.4; 120.4+35.7 = 156.1; 156.1+110.1 = 266.2; 266.2+54.1 = 320.3; 320.3+99 = 419.3 \)

Calculate \( \sum xy \):
\( 48.4\times38=1839.2 \); \( 42.6\times7.5 = 319.5 \); \( 56.5\times(-29.2)=-1649.8 \); \( 29.2\times94.5 = 2759.4 \); \( 46.1\times9.6 = 442.56 \); \( 38\times35.7 = 1356.6 \); \( 10\times110.1 = 1101 \); \( 70\times54.1 = 3787 \); \( 35\times99 = 3465 \)

\( \sum xy=1839.2+319.5-1649.8 + 2759.4+442.56+1356.6+1101+3787+3465 \)
\( 1839.2+319.5 = 2158.7; 2158.7-1649.8 = 508.9; 508.9+2759.4 = 3268.3; 3268.3+442.56 = 3710.86; 3710.86+1356.6 = 5067.46; 5067.46+1101 = 6168.46; 6168.46+3787 = 9955.46; 9955.46+3465 = 13420.46 \)

Calculate \( \sum x^2 \):
\( 48.4^2=2342.56 \); \( 42.6^2 = 1814.76 \); \( 56.5^2 = 3192.25 \); \( 29.2^2 = 852.64 \); \( 46.1^2 = 2125.21 \); \( 38^2 = 1444 \); \( 10^2 = 100 \); \( 70^2 = 4900 \); \( 35^2 = 1225 \)

\( \sum x^2=2342.56+1814.76+3192.25+852.64+2125.21+1444+100+4900+1225 \)
\( 2342.56+1814.76 = 4157.32; 4157.32+3192.25 = 7349.57; 7349.57+852.64 = 8202.21; 8202.21+2125.21 = 10327.42; 10327.42+1444 = 11771.42; 11771.42+100 = 11871.42; 11871.42+4900 = 16771.42; 16771.42+1225 = 17996.42 \)

Step2: Calculate slope \( m \)

The formula for the slope \( m \) of the regression line is:
\( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2} \)

Substitute the values:
\( n = 9 \), \( \sum xy = 13420.46 \), \( \sum x = 375.8 \), \( \sum y = 419.3 \), \( \sum x^2=17996.42 \)

Numerator: \( 9\times13420.46-375.8\times419.3 \)
\( 9\times13420.46 = 120784.14 \)
\( 375.8\times419.3 \approx375.8\times400=150320; 375.8\times19.3\approx7252.94; \) so \( 150320 + 7252.94=157572.94 \)
Numerator: \( 120784.14-157572.94=-36788.8 \)

Denominator: \( 9\times17996.42-(375.8)^2 \)
\( 9\times17996.42 = 161967.78 \)
\( (375.8)^2 = 375.8\times375.8 \approx141225.64 \)
Denominator: \( 161967.78 - 141225.64 = 20742.14 \)

So, \( m=\frac{-36788.8}{20742.14}\approx - 1.77 \) (rounded to two decimal places)

Step3: Calculate y-intercept \( b \)

The formula for the y-intercept \( b \) is:
\( b=\frac{\sum y - m\sum x}{n} \)

Substitute the values:
\( \sum y = 419.3 \), \( m\approx - 1.77 \), \( \sum x = 375.8 \), \( n = 9 \)

\( m\sum x\approx-1.77\times375.8\approx - 665.17 \)

\( \sum y - m\sum x\approx419.3+665.17 = 1084.47 \)

\( b=\frac{1084.47}{9}\approx120.497\approx120.50 \) (rounded to two decimal places)

Step4: Write the regression equation

The slope - intercept form of a line is \( y = mx + b \). Substituting \( m\approx - 1.77 \) and \( b\approx120.50 \), we get:
\( y=-1.77x + 120.50 \)

Answer:

\( y = -1.77x + 120.50 \)