Question

1. the data in the table below shows the number of passengers and number of suitcases on various airplanes.

passengers suitcases
75 159
92 180
115 239
128 272
143 290
154 310
178 357
200 405
a. find the line of best fit:____suitcases
b. estimate the number of suitcases on a flight carrying 340 people.

Explanation:

Step1: Calculate the mean of passengers ($x$) and suitcases ($y$)

Let \(x\) be the number of passengers and \(y\) be the number of suitcases.
\(\bar{x}=\frac{75 + 92+115+128+143+154+178+200}{8}=\frac{1085}{8}=135.625\)
\(\bar{y}=\frac{159+180 + 239+272+290+310+357+405}{8}=\frac{2212}{8}=276.5\)

Step2: Calculate the slope \(m\)

$$m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}$$

\(\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=(75 - 135.625)(159 - 276.5)+(92-135.625)(180 - 276.5)+(115 - 135.625)(239 - 276.5)+(128-135.625)(272 - 276.5)+(143 - 135.625)(290 - 276.5)+(154-135.625)(310 - 276.5)+(178 - 135.625)(357 - 276.5)+(200-135.625)(405 - 276.5)\)

$$=( - 60.625)(- 117.5)+(-43.625)(-96.5)+(-20.625)(-37.5)+(-7.625)(-4.5)+(7.375)(13.5)+(18.375)(33.5)+(42.375)(80.5)+(64.375)(128.5)$$

$$=7123.4375 + 4219.8125+773.4375 + 34.3125+99.5625+615.5625+3411.1875+8272.2875$$

$$=24559.625$$

\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=(75 - 135.625)^{2}+(92 - 135.625)^{2}+(115 - 135.625)^{2}+(128 - 135.625)^{2}+(143 - 135.625)^{2}+(154 - 135.625)^{2}+(178 - 135.625)^{2}+(200 - 135.625)^{2}\)

$$=(-60.625)^{2}+(-43.625)^{2}+(-20.625)^{2}+(-7.625)^{2}+(7.375)^{2}+(18.375)^{2}+(42.375)^{2}+(64.375)^{2}$$

$$=3675.390625+1903.140625 + 425.390625+58.140625+54.390625+337.640625+1795.640625+4144.140625$$

$$=12393.875$$

\(m=\frac{24559.625}{12393.875}\approx2\)

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

Using the formula \(y=mx + b\), substitute \(m = 2\), \(x=\bar{x}=135.625\) and \(y=\bar{y}=276.5\)
\(276.5=2\times135.625 + b\)
\(276.5 = 271.25+b\)
\(b=276.5 - 271.25=5.25\)

The line of best fit is \(y = 2x+5.25\)

Step4: Estimate for \(x = 340\)

Substitute \(x = 340\) into \(y=2x + 5.25\)
\(y=2\times340+5.25=680 + 5.25=685.25\approx685\)

Answer:

a. The line of best fit is \(y = 2x+5.25\)
b. The estimated number of suitcases is \(685\)