Question

find the equation for the least squares regression line of the data described below. sandeep wants to determine how much bottled water he should stock in his store on any given day. he believes that sales of bottled water are much higher on hotter days. to test this hypothesis, sandeep tracked his water bottle sales over several days. he recorded the high temperature (in celsius), x, and the number of bottles sold, y, each day. temperature (in celsius) | bottles: 31.12 | 179; 31.30 | 162; 33.77 | 194; 35.95 | 196; 38.63 | 191. round your answers to the nearest thousandth. y = x +

Explanation:

Step1: Identify the data points

We have the following data points \((x, y)\) where \(x\) is temperature (in Celsius) and \(y\) is number of bottles sold:
\((31.12, 179)\), \((31.30, 162)\), \((33.77, 194)\), \((35.95, 196)\), \((38.63, 191)\)

Step2: Calculate the necessary sums

First, we calculate \(n = 5\) (number of data points).
Now, calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^{2}\):

• \(\sum x=31.12 + 31.30+33.77 + 35.95+38.63=170.77\)
• \(\sum y=179 + 162+194 + 196+191 = 922\)
• \(\sum xy=(31.12\times179)+(31.30\times162)+(33.77\times194)+(35.95\times196)+(38.63\times191)\)

\(=31.12\times179 = 5570.48\)
\(=31.30\times162 = 5070.6\)
\(=33.77\times194 = 6551.38\)
\(=35.95\times196 = 7046.2\)
\(=38.63\times191 = 7378.33\)
\(\sum xy=5570.48+5070.6 + 6551.38+7046.2+7378.33=31616.99\)

• \(\sum x^{2}=(31.12)^{2}+(31.30)^{2}+(33.77)^{2}+(35.95)^{2}+(38.63)^{2}\)

\(=31.12^{2}=968.4544\)
\(=31.30^{2}=979.69\)
\(=33.77^{2}=1139.4129\)
\(=35.95^{2}=1292.4025\)
\(=38.63^{2}=1492.2769\)
\(\sum x^{2}=968.4544 + 979.69+1139.4129+1292.4025+1492.2769 = 5872.2367\)

Step3: Calculate the slope \(m\)

The formula for the slope \(m\) of the least - squares regression line \(y = mx + b\) is:
\(m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\)
Substitute the values:
\(n = 5\), \(\sum xy = 31616.99\), \(\sum x=170.77\), \(\sum y = 922\), \(\sum x^{2}=5872.2367\)
\(n\sum xy-\sum x\sum y=5\times31616.99-170.77\times922\)
\(=158084.95-157450.94 = 634.01\)
\(n\sum x^{2}-(\sum x)^{2}=5\times5872.2367-(170.77)^{2}\)
\(=29361.1835 - 29162.3929=198.7906\)
\(m=\frac{634.01}{198.7906}\approx3.190\)

Step4: Calculate the y - intercept \(b\)

The formula for the y - intercept \(b\) is:
\(b=\frac{\sum y - m\sum x}{n}\)
Substitute the values:
\(\sum y = 922\), \(m\approx3.190\), \(\sum x = 170.77\), \(n = 5\)
\(m\sum x=3.190\times170.77\approx544.76\)
\(\sum y - m\sum x=922 - 544.76 = 377.24\)
\(b=\frac{377.24}{5}=75.448\)

Answer:

The equation of the least - squares regression line is \(y = 3.190x+75.448\) (rounded to the nearest thousandth)