Question

use the given data to find the equation of the regression line. examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.

x
5
6
9
13
10
12
11
8
14
7
4

y
5.54
7.38
10.78
10.34
11.20
10.98
11.26
10.00
9.34
8.86
3.34

\\( \hat { y } = \square + \square x \\) (round to two decimal places as needed.)

Explanation:

Step1: Organize the data

We have the following data pairs \((x, y)\):
\((5, 5.54)\), \((6, 7.38)\), \((9, 10.78)\), \((13, 10.34)\), \((10, 11.20)\), \((12, 10.98)\), \((11, 11.26)\), \((8, 10.00)\), \((14, 9.34)\), \((7, 8.86)\), \((4, 3.34)\)

First, we calculate the necessary sums:
Let \(n\) be the number of data points. Here, \(n = 11\)

Calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^{2}\)

\(\sum x=5 + 6+9 + 13+10+12+11+8+14+7+4\)
\(=5 + 6=11\); \(11+9 = 20\); \(20 + 13=33\); \(33+10 = 43\); \(43+12 = 55\); \(55+11 = 66\); \(66+8 = 74\); \(74+14 = 88\); \(88+7 = 95\); \(95 + 4=99\)

\(\sum y=5.54+7.38+10.78+10.34+11.20+10.98+11.26+10.00+9.34+8.86+3.34\)
\(5.54+7.38 = 12.92\); \(12.92+10.78 = 23.7\); \(23.7+10.34 = 34.04\); \(34.04+11.20 = 45.24\); \(45.24+10.98 = 56.22\); \(56.22+11.26 = 67.48\); \(67.48+10.00 = 77.48\); \(77.48+9.34 = 86.82\); \(86.82+8.86 = 95.68\); \(95.68+3.34 = 99.02\)

\(\sum xy=(5\times5.54)+(6\times7.38)+(9\times10.78)+(13\times10.34)+(10\times11.20)+(12\times10.98)+(11\times11.26)+(8\times10.00)+(14\times9.34)+(7\times8.86)+(4\times3.34)\)
\(27.7+44.28 = 71.98\); \(71.98+97.02 = 169\); \(169+134.42 = 303.42\); \(303.42+112 = 415.42\); \(415.42+131.76 = 547.18\); \(547.18+123.86 = 671.04\); \(671.04+80 = 751.04\); \(751.04+130.76 = 881.8\); \(881.8+62.02 = 943.82\); \(943.82+13.36 = 957.18\)

\(\sum x^{2}=5^{2}+6^{2}+9^{2}+13^{2}+10^{2}+12^{2}+11^{2}+8^{2}+14^{2}+7^{2}+4^{2}\)
\(25 + 36=61\); \(61+81 = 142\); \(142+169 = 311\); \(311+100 = 411\); \(411+144 = 555\); \(555+121 = 676\); \(676+64 = 740\); \(740+196 = 936\); \(936+49 = 985\); \(985+16 = 1001\)

Step2: Calculate the 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:
\(n = 11\), \(\sum xy=957.18\), \(\sum x = 99\), \(\sum y=99.02\), \(\sum x^{2}=1001\)

\(n\sum xy=11\times957.18 = 10528.98\)

\(\sum x\sum y=99\times99.02=9802.98\)

\(n\sum x^{2}=11\times1001 = 11011\)

\((\sum x)^{2}=99^{2}=9801\)

\(m=\frac{10528.98 - 9802.98}{11011-9801}=\frac{726}{1210}\approx0.60\)

Step3: Calculate the y-intercept \(b\)

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

Substitute the values:
\(\sum y = 99.02\), \(m\approx0.60\), \(\sum x = 99\), \(n = 11\)

\(m\sum x=0.60\times99 = 59.4\)

\(\sum y - m\sum x=99.02 - 59.4=39.62\)

\(b=\frac{39.62}{11}\approx3.60\)

Answer:

The equation of the regression line is \(y = 0.60x+3.60\) (rounded to two decimal places)