Question

problem 5
mrs. jones - johnson has a t - shirt printing business. the table shows the number of t - shirts mrs. jones - johnson has sold each year since starting her business in 2010

years|number of t - shirts
2010|50
2011|75
2012|175
2013|125
2014|250
2015|300
2016|375

regression equation:
predict how many t - shirts mrs. jones - johnson will sell in 2024.
predict how many t - shirts mrs. jones - johnson will sell in 2018.

Explanation:

Step1: Set up data

Let \(x = 0\) represent the year 2010. So for 2010 \(x = 0\), for 2011 \(x=1\), and so on. The data points are \((0,50),(1,75),(2,175),(3,125),(4,250),(5,300),(6,375)\).

Step2: Use linear - regression formula

The linear - regression equation is of the form \(y = mx + b\), where \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\frac{\sum_{i = 1}^{n}y_i - m\sum_{i = 1}^{n}x_i}{n}\).
First, calculate the necessary sums:
\(\sum_{i = 0}^{6}x_i=0 + 1+2 + 3+4+5+6=\frac{6\times(6 + 1)}{2}=21\)
\(\sum_{i = 0}^{6}y_i=50 + 75+175+125+250+300+375 = 1350\)
\(\sum_{i = 0}^{6}x_i^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=\frac{6\times(6 + 1)\times(2\times6 + 1)}{6}=91\)
\(\sum_{i = 0}^{6}x_iy_i=0\times50+1\times75+2\times175+3\times125+4\times250+5\times300+6\times375\)
\(=75 + 350+375+1000+1500+2250=5550\)
\(n = 7\)
\(m=\frac{7\times5550-21\times1350}{7\times91 - 21^{2}}=\frac{38850-28350}{637 - 441}=\frac{10500}{196}\approx53.57\)
\(b=\frac{1350-53.57\times21}{7}=\frac{1350 - 1125}{7}=\frac{225}{7}\approx32.14\)
The regression equation is \(y = 53.57x+32.14\)

Step3: Predict for 2018

For 2018, \(x = 8\) (since 2010 is \(x = 0\)).
\(y=53.57\times8+32.14=428.56+32.14 = 460.7\approx461\)

Step4: Predict for 2024

For 2024, \(x = 14\) (since 2010 is \(x = 0\)).
\(y=53.57\times14+32.14=749.98+32.14=782.12\approx782\)

Answer:

Regression Equation: \(y = 53.57x+32.14\)
Number of t - shirts sold in 2018: 461
Number of t - shirts sold in 2024: 782