Question

dante collected the following data from a recent survey:

x	2	3	4	5	6	7	9
y	3	4	3	5	7	7	8

complete the parts below to help dante find a relationship between x and y for his dataset.
(a) to see if there are any trends in the data, dante wants to create a scatter plot. he plotted the first four data points. plot the remaining three data points.
(b) the data seems to have a linear trend. so, dante wants to find a linear equation that could approximately model the data. select \compute\ to generate the values used for a linear regression model. write the values in your answer to 3 decimal places.

x	2	3	4	5	6	7	9
y	3	4	3	5	7	7	8

lin reg: y = a x + b
a =
b =
r² =
r =
the linear equation that best fits the data is y =
(c) use the equation from part (b) to help dante predict the value of y when x = 1. write your answer to 3 decimal places.
y =

Explanation:

Step 1: Calculate necessary sums for linear regression

First, we list the data points: \((2, 3)\), \((3, 4)\), \((4, 3)\), \((5, 5)\), \((6, 7)\), \((7, 7)\), \((9, 8)\).

The number of data points \(n = 7\).

Calculate \(\sum x\): \(2 + 3 + 4 + 5 + 6 + 7 + 9 = 36\)

Calculate \(\sum y\): \(3 + 4 + 3 + 5 + 7 + 7 + 8 = 37\)

Calculate \(\sum xy\): \((2\times3)+(3\times4)+(4\times3)+(5\times5)+(6\times7)+(7\times7)+(9\times8)= 6 + 12 + 12 + 25 + 42 + 49 + 72 = 218\)

Calculate \(\sum x^2\): \(2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 9^2 = 4 + 9 + 16 + 25 + 36 + 49 + 81 = 220\)

Step 2: Calculate the slope \(a\)

The formula for the slope \(a\) in linear regression is:

$$ a=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2} $$

Substitute the values:

$$ a=\frac{7\times218 - 36\times37}{7\times220 - 36^2}=\frac{1526 - 1332}{1540 - 1296}=\frac{194}{244}\approx0.795 $$

Step 3: Calculate the y-intercept \(b\)

The formula for the y-intercept \(b\) is:

$$ b=\frac{\sum y - a\sum x}{n} $$

Substitute the values:

$$ b=\frac{37 - 0.795\times36}{7}=\frac{37 - 28.62}{7}=\frac{8.38}{7}\approx1.197 $$

Step 4: Calculate the correlation coefficient \(r\) and \(r^2\)

The formula for \(r\) is:

$$ r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

First, calculate \(\sum y^2\): \(3^2 + 4^2 + 3^2 + 5^2 + 7^2 + 7^2 + 8^2 = 9 + 16 + 9 + 25 + 49 + 49 + 64 = 221\)

Then,

$$ r=\frac{7\times218 - 36\times37}{\sqrt{[7\times220 - 36^2][7\times221 - 37^2]}}=\frac{194}{\sqrt{244\times(1547 - 1369)}}=\frac{194}{\sqrt{244\times178}}=\frac{194}{\sqrt{43432}}\approx\frac{194}{208.4}\approx0.930 $$

Then \(r^2=(0.930)^2\approx0.865\)

Step 5: Predict \(y\) when \(x = 1\)

Using the linear equation \(y = ax + b\), substitute \(x = 1\), \(a\approx0.795\) and \(b\approx1.197\):

$$ y=0.795\times1 + 1.197\approx1.992 $$

Answer:

Part (b)

\(a\approx0.795\), \(b\approx1.197\), \(r^2\approx0.865\), \(r\approx0.930\)

The linear equation is \(y = 0.795x + 1.197\)

Part (c)

When \(x = 1\), \(y\approx1.992\)