Question

problem 2:
the table below shows the average annual salary of the workers at the samsan corporation over time since 2000.
let ( x ) represent the number of years since 2000 with ( x = 0 ) representing the year 2000. let ( y ) represent the annual salary of the workers.
complete the slope - intercept form of the equation for the line of best fit below with the appropriate value for the slope. round to the nearest whole number.

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

Let's first find the \(x\) - values (years since 2000). For 2000, \(x = 0\); for 2008, \(x=8\); for 2010, \(x = 10\); for 2011, \(x=11\); for 2013, \(x = 13\); for 2015, \(x=15\); for 2017, \(x = 17\); for 2018, \(x=18\); for 2021, \(x=21\).
The \(x\) - values are \(x=\{0,8,10,11,13,15,17,18,21\}\)
The sum of \(x\) - values \(\sum x=0 + 8+10+11+13+15+17+18+21=113\)
The number of data points \(n = 9\)
The mean of \(x\), \(\bar{x}=\frac{\sum x}{n}=\frac{113}{9}\approx12.56\)

The \(y\) - values are \(y =\{16570,31226,33554,40538,42866,49850,52178,56834,60076\}\)
The sum of \(y\) - values \(\sum y=16570+31226+33554+40538+42866+49850+52178+56834+60076 = 383692\)
The mean of \(y\), \(\bar{y}=\frac{\sum y}{n}=\frac{383692}{9}\approx42632.44\)

Step2: Calculate the numerator and denominator for the slope formula

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

First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) for each \(i\):

• For \(x = 0,y = 16570\): \((0 - 12.56)(16570-42632.44)=(- 12.56)\times(-26062.44)=327344.2464\)
• For \(x = 8,y = 31226\): \((8 - 12.56)(31226 - 42632.44)=(-4.56)\times(-11406.44)=52013.3664\)
• For \(x = 10,y = 33554\): \((10 - 12.56)(33554-42632.44)=(-2.56)\times(-9078.44)=23240.8064\)
• For \(x = 11,y = 40538\): \((11 - 12.56)(40538-42632.44)=(-1.56)\times(-2094.44)=3267.3264\)
• For \(x = 13,y = 42866\): \((13 - 12.56)(42866-42632.44)=(0.44)\times(233.56)=102.7664\)
• For \(x = 15,y = 49850\): \((15 - 12.56)(49850-42632.44)=(2.44)\times(7217.56)=17610.8464\)
• For \(x = 17,y = 52178\): \((17 - 12.56)(52178-42632.44)=(4.44)\times(9545.56)=42382.2864\)
• For \(x = 18,y = 56834\): \((18 - 12.56)(56834-42632.44)=(5.44)\times(14201.56)=77256.4864\)
• For \(x = 21,y = 60076\): \((21 - 12.56)(60076-42632.44)=(8.44)\times(17443.56)=147223.6464\)

The sum of \((x_{i}-\bar{x})(y_{i}-\bar{y})\) is \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=327344.2464+52013.3664+23240.8064+3267.3264+102.7664+17610.8464+42382.2864+77256.4864+147223.6464 = 690442.86\)

Next, calculate \((x_{i}-\bar{x})^{2}\) for each \(i\):

• For \(x = 0\): \((0 - 12.56)^{2}=157.7536\)
• For \(x = 8\): \((8 - 12.56)^{2}=20.7936\)
• For \(x = 10\): \((10 - 12.56)^{2}=6.5536\)
• For \(x = 11\): \((11 - 12.56)^{2}=2.4336\)
• For \(x = 13\): \((13 - 12.56)^{2}=0.1936\)
• For \(x = 15\): \((15 - 12.56)^{2}=5.9536\)
• For \(x = 17\): \((17 - 12.56)^{2}=19.7136\)
• For \(x = 18\): \((18 - 12.56)^{2}=29.5936\)
• For \(x = 21\): \((21 - 12.56)^{2}=71.2336\)

The sum of \((x_{i}-\bar{x})^{2}\) is \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=157.7536+20.7936+6.5536+2.4336+0.1936+5.9536+19.7136+29.5936+71.2336=314.222\)

Step3: Calculate the slope

\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}=\frac{690442.86}{314.222}\approx2200\)

Answer:

The slope \(m\approx2200\)