Question

what is the slope of the least - squares regression line for these data? carry your intermediate computations to decimal places and round your answer to at least two decimal places. (if necessary, consult a list of formulas.

Explanation:

Step1: Calculate the necessary sums

Let \(n = 14\) (number of data points).
\(\sum x=62.8 + 56.5+110.2+54.5+74.9+54.4+49.4+41.3+133.4+41.9+86.1+34.7+106.9+66.8=978.8\)
\(\sum y=28.52 + 33.09+32.72+20.74+32.35+18.52+16.30+23.70+42.84+26.79+43.70+13.21+29.01+20.25 = 382.24\)
\(\sum xy=1791.056+1869.585+3605.744+1130.33+2423.015+1007.488+805.22+978.81+5714.856+1122.501+3762.57+458.387+3101.169+1352.7=27123.438\)
\(\sum x^{2}=62.8^{2}+56.5^{2}+110.2^{2}+54.5^{2}+74.9^{2}+54.4^{2}+49.4^{2}+41.3^{2}+133.4^{2}+41.9^{2}+86.1^{2}+34.7^{2}+106.9^{2}+66.8^{2}\)
\(=3943.84+3192.25+12144.04+2970.25+5610.01+2959.36+2440.36+1705.69+17795.56+1755.61+7413.21+1204.09+11427.61+4462.24 = 76024.12\)

Step2: Use the formula for the slope \(b\) of the least - squares regression line

The formula for the slope \(b=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\)
Substitute \(n = 14\), \(\sum x = 978.8\), \(\sum y=382.24\), \(\sum xy = 27123.438\), \(\sum x^{2}=76024.12\) into the formula:

$$
LATEXBLOCK0
$$

First, calculate \(978.8\times382.24=(1000 - 21.2)\times382.24=1000\times382.24-21.2\times382.24=382240-8103.488 = 374136.512\)
Then \(379728.132-374136.512 = 5591.62\)
And \(1064337.68 - 958049.44=106288.24\)
\(b=\frac{5591.62}{106288.24}\approx0.05\)

Answer:

\(0.05\)