Question

given the points (1, 2), (2, 3), (3, 5), (4, 4), and (5, 6), estimate the equation of a line of best fit. please answer in the form of an equation y=mx+b adding no spaces in your answer question 2 what does each point on a scatterplot represent? a pair of values for two variables a single value for one variable a line of best fit a category label

Explanation:

Question 1:

Step1: Calculate the mean of x and y

First, we find the mean of the x - values: \(x=\{1,2,3,4,5\}\), so \(\bar{x}=\frac{1 + 2+3 + 4+5}{5}=\frac{15}{5} = 3\)
Then, the mean of the y - values: \(y=\{2,3,5,4,6\}\), so \(\bar{y}=\frac{2 + 3+5 + 4+6}{5}=\frac{20}{5}=4\)

Step2: Calculate the slope (m)

We use the formula for the slope of the line of best fit \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

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

• For \((1,2)\): \((1 - 3)(2 - 4)=(-2)\times(-2) = 4\)
• For \((2,3)\): \((2 - 3)(3 - 4)=(-1)\times(-1)=1\)
• For \((3,5)\): \((3 - 3)(5 - 4)=0\times1 = 0\)
• For \((4,4)\): \((4 - 3)(4 - 4)=1\times0 = 0\)
• For \((5,6)\): \((5 - 3)(6 - 4)=2\times2 = 4\)

Sum of \((x_{i}-\bar{x})(y_{i}-\bar{y})\): \(4 + 1+0 + 0+4=9\)

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

• For \((1,2)\): \((1 - 3)^{2}=(-2)^{2}=4\)
• For \((2,3)\): \((2 - 3)^{2}=(-1)^{2}=1\)
• For \((3,5)\): \((3 - 3)^{2}=0^{2}=0\)
• For \((4,4)\): \((4 - 3)^{2}=1^{2}=1\)
• For \((5,6)\): \((5 - 3)^{2}=2^{2}=4\)

Sum of \((x_{i}-\bar{x})^{2}\): \(4 + 1+0 + 1+4 = 10\)

So, \(m=\frac{9}{10}=0.9\)

Step3: Calculate the y - intercept (b)

We use the formula \(y=\bar{y}-m\bar{x}\)
Substitute \(\bar{y} = 4\), \(m = 0.9\) and \(\bar{x}=3\) into the formula:
\(b=4-0.9\times3=4 - 2.7 = 1.3\)

A scatterplot is a graph that shows the relationship between two variables. Each point on the scatterplot is plotted using a pair of values, one for each of the two variables (usually on the x - axis and y - axis). A single value for one variable would be a point on a number line, not a scatterplot. A line of best fit is a line drawn through the scatterplot, not a point. A category label is used for categorical data in other types of graphs, not for the points in a scatterplot.

Answer:

\(y = 0.9x+1.3\) (or if we use a more approximate method by looking at the trend, we can also get \(y=x + 1\) as a rough estimate. But the calculated one is \(y = 0.9x+1.3\))

Question 2: