Question

dante collected the following data from a science experiment.

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.
(there is a scatter plot with some points plotted and a plotting tool.)
(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 to 3 decimal places.

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

lin reg: ( y = ax + b )
( a = )
( b = )
( r^2 = )
( 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 need to find the sums of \( x \), \( y \), \( xy \), and \( x^2 \).

Given data:
\( x: [2, 3, 4, 5, 6, 7, 9] \)
\( y: [3, 4, 3, 5, 7, 7, 8] \)

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

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

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

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

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

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:
\( a = \frac{7\times218 - 36\times37}{7\times220 - 36^2} \)
\( = \frac{1526 - 1332}{1540 - 1296} \)
\( = \frac{194}{244} \approx 0.795 \)

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:
\( b = \frac{37 - 0.795\times36}{7} \)
\( = \frac{37 - 28.62}{7} \)
\( = \frac{8.38}{7} \approx 1.197 \)

Step 4: Write the linear equation

The linear equation is \( y = ax + b \), so substituting \( a \) and \( b \):
\( y = 0.795x + 1.197 \)

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

Substitute \( x = 1 \) into the equation:
\( y = 0.795\times1 + 1.197 = 0.795 + 1.197 = 1.992 \approx 1.992 \)

Answer:

Part (b) Linear Equation: \( y = 0.795x + 1.197 \)

Part (c) Predicted \( y \) when \( x = 1 \): \( 1.992 \)