Question

the average number of cars entering a shopping center parking lot is collected and shown in the table below.
predict how many cars will enter the parking lot at approximately 300 seconds. round to the nearest car.
cars: type your answer...

Explanation:

Step1: Find the regression equation

We can use a statistical software or a graphing calculator to find the regression equation. Assuming we use linear regression (since the problem doesn't specify another type), let \(x\) be the time (in seconds) and \(y\) be the number of cars.
If we input the data \((x_1,y_1)=(20,70),(x_2 = 60,y_2=178),(x_3=100,y_3 = 260),(x_4=140,y_4=301),(x_5=180,y_5=284),(x_6=220,y_6=260),(x_7=240,y_7=195)\) into a calculator or software for linear regression \(y=ax + b\).
The formula for the slope \(a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i=1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i=1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\frac{\sum_{i=1}^{n}y_i - a\sum_{i=1}^{n}x_i}{n}\), where \(n = 7\).
\(\sum_{i=1}^{7}x_i=20 + 60+100+140+180+220+240=960\)
\(\sum_{i=1}^{7}y_i=70 + 178+260+301+284+260+195=1548\)
\(\sum_{i=1}^{7}x_iy_i=20\times70+60\times178 + 100\times260+140\times301+180\times284+220\times260+240\times195\)
\(=1400+10680+26000+42140+51120+57200+46800=235340\)
\(\sum_{i=1}^{7}x_i^{2}=20^{2}+60^{2}+100^{2}+140^{2}+180^{2}+220^{2}+240^{2}\)
\(=400 + 3600+10000+19600+32400+48400+57600=172000\)
\(a=\frac{7\times235340-960\times1548}{7\times172000 - 960^{2}}\)
\(=\frac{1647380-1486080}{1204000 - 921600}=\frac{161300}{282400}\approx0.571\)
\(b=\frac{1548-0.571\times960}{7}=\frac{1548 - 548.16}{7}=\frac{999.84}{7}\approx142.83\)
So the regression equation is \(y = 0.571x+142.83\)

Step2: Predict for \(x = 300\)

Substitute \(x = 300\) into the regression equation \(y=0.571\times300+142.83\)
\(y=171.3+142.83=314.13\approx314\)

Answer:

314