Question

question 10
based on the data shown below, calculate the regression line (each value to two decimal places)
y =
\bx +

x	y
4	20.46
5	20.55
6	20.84
7	21.53
8	20.12
9	23.51
10	22.9
11	21.09
12	23.18
13	21.17
14	23.06
15	24.65
16	24.24
17	22.83
18	23.22
19	26.21

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Explanation:

Step1: Calculate necessary sums

First, we need to calculate \( n \) (number of data points), \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \).

Given data points:
\( x: 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 \)
\( y: 20.46,20.55,20.84,21.53,20.12,23.51,22.9,21.09,23.18,21.17,23.06,24.65,24.24,22.83,23.22,26.21 \)

\( n = 16 \)

\( \sum x = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 \)
\( = \frac{16(4 + 19)}{2} = 16\times\frac{23}{2} = 184 \)

\( \sum y = 20.46 + 20.55 + 20.84 + 21.53 + 20.12 + 23.51 + 22.9 + 21.09 + 23.18 + 21.17 + 23.06 + 24.65 + 24.24 + 22.83 + 23.22 + 26.21 \)
Let's calculate step by step:
20.46 + 20.55 = 41.01; 41.01 + 20.84 = 61.85; 61.85 + 21.53 = 83.38; 83.38 + 20.12 = 103.5; 103.5 + 23.51 = 127.01; 127.01 + 22.9 = 149.91; 149.91 + 21.09 = 171; 171 + 23.18 = 194.18; 194.18 + 21.17 = 215.35; 215.35 + 23.06 = 238.41; 238.41 + 24.65 = 263.06; 263.06 + 24.24 = 287.3; 287.3 + 22.83 = 310.13; 310.13 + 23.22 = 333.35; 333.35 + 26.21 = 359.56
So, \( \sum y = 359.56 \)

\( \sum xy \):
Calculate each \( x \times y \) and sum:
4×20.46 = 81.84; 5×20.55 = 102.75; 6×20.84 = 125.04; 7×21.53 = 150.71; 8×20.12 = 160.96; 9×23.51 = 211.59; 10×22.9 = 229; 11×21.09 = 231.99; 12×23.18 = 278.16; 13×21.17 = 275.21; 14×23.06 = 322.84; 15×24.65 = 369.75; 16×24.24 = 387.84; 17×22.83 = 388.11; 18×23.22 = 417.96; 19×26.21 = 497.99
Now sum these:
81.84 + 102.75 = 184.59; +125.04 = 309.63; +150.71 = 460.34; +160.96 = 621.3; +211.59 = 832.89; +229 = 1061.89; +231.99 = 1293.88; +278.16 = 1572.04; +275.21 = 1847.25; +322.84 = 2170.09; +369.75 = 2539.84; +387.84 = 2927.68; +388.11 = 3315.79; +417.96 = 3733.75; +497.99 = 4231.74
So, \( \sum xy = 4231.74 \)

\( \sum x^2 \):
Calculate each \( x^2 \) and sum:
\( 4^2 = 16 \); \( 5^2 = 25 \); \( 6^2 = 36 \); \( 7^2 = 49 \); \( 8^2 = 64 \); \( 9^2 = 81 \); \( 10^2 = 100 \); \( 11^2 = 121 \); \( 12^2 = 144 \); \( 13^2 = 169 \); \( 14^2 = 196 \); \( 15^2 = 225 \); \( 16^2 = 256 \); \( 17^2 = 289 \); \( 18^2 = 324 \); \( 19^2 = 361 \)
Sum: 16 + 25 = 41; +36 = 77; +49 = 126; +64 = 190; +81 = 271; +100 = 371; +121 = 492; +144 = 636; +169 = 805; +196 = 1001; +225 = 1226; +256 = 1482; +289 = 1771; +324 = 2095; +361 = 2456
So, \( \sum x^2 = 2456 \)

Step2: Calculate slope (\( m \)) and intercept (\( b \))

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

First, calculate \( m \):
\( n\sum xy = 16\times4231.74 = 67707.84 \)
\( \sum x \sum y = 184\times359.56 = 184\times359.56 \)
Calculate 180×359.56 = 64720.8; 4×359.56 = 1438.24; total = 64720.8 + 1438.24 = 66159.04
\( n\sum x^2 = 16\times2456 = 39296 \)
\( (\sum x)^2 = 184^2 = 33856 \)

So, numerator for \( m \): \( 67707.84 - 66159.04 = 1548.8 \)
Denominator for \( m \): \( 39296 - 33856 = 5440 \)
\( m = \frac{1548.8}{5440} \approx 0.28 \) (rounded to two decimal places)

Now calculate \( b \):
\( m\sum x = 0.28\times184 = 51.52 \)
\( \sum y - m\sum x = 359.56 - 51.52 = 308.04 \)
\( b = \frac{308.04}{16} \approx 19.25 \) (rounded to two decimal places)

Answer:

\( y = 0.28x + 19.25 \)