Question

calculator active. answer the questions pertaining to the given data.

1. a. plot the following data on both graphs below.

x	2	4	5	8	9
y	7	15	23	77	115

graphs
b. find a regression equation for the above data.
c. take the log of both sides and use log rules to create a linear function.
d. complete the table to find log y.

x	2	4	5	8	9
y	7	15	23	77	115
log y

e. find a linear regression equation for (x, log y).

1. a. plot the following data on both graphs below.

x	1	3	4	5	7
y	65.0	32.5	16.3	8.2	1.0

graphs

Explanation:

Part 4d: Complete the table to find \(\boldsymbol{\log y}\) (assuming base 10 logarithm)

Step 1: Calculate \(\log(7)\)

Using a calculator, \(\log(7) \approx 0.8451\)

Step 2: Calculate \(\log(15)\)

Using a calculator, \(\log(15) \approx 1.1761\)

Step 3: Calculate \(\log(23)\)

Using a calculator, \(\log(23) \approx 1.3617\)

Step 4: Calculate \(\log(77)\)

Using a calculator, \(\log(77) \approx 1.8865\)

Step 5: Calculate \(\log(115)\)

Using a calculator, \(\log(115) \approx 2.0607\)

Step 1: List the data points \((x, \log y)\)

From part 4d, the points are \((2, 0.8451)\), \((4, 1.1761)\), \((5, 1.3617)\), \((8, 1.8865)\), \((9, 2.0607)\)

Step 2: Calculate the mean of \(x\) (\(\bar{x}\)) and mean of \(\log y\) (\(\bar{\log y}\))

\[
\bar{x}=\frac{2 + 4+5 + 8+9}{5}=\frac{28}{5} = 5.6
\]
\[
\bar{\log y}=\frac{0.8451+1.1761 + 1.3617+1.8865+2.0607}{5}=\frac{7.3301}{5}=1.46602
\]

Step 3: Calculate the slope \(m\) of the linear regression line

The formula for slope \(m\) is:
\[
m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(\log y_i-\bar{\log y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}
\]
First, calculate \((x_i-\bar{x})\) and \((\log y_i-\bar{\log y})\) for each point:

• For \((2, 0.8451)\): \(x - \bar{x}=2 - 5.6=- 3.6\), \(\log y-\bar{\log y}=0.8451 - 1.46602=-0.62092\), product \(=(-3.6)\times(-0.62092) = 2.235312\), \((x - \bar{x})^2=(-3.6)^2 = 12.96\)
• For \((4, 1.1761)\): \(x - \bar{x}=4 - 5.6=-1.6\), \(\log y-\bar{\log y}=1.1761 - 1.46602=-0.28992\), product \(=(-1.6)\times(-0.28992)=0.463872\), \((x - \bar{x})^2=(-1.6)^2 = 2.56\)
• For \((5, 1.3617)\): \(x - \bar{x}=5 - 5.6=-0.6\), \(\log y-\bar{\log y}=1.3617 - 1.46602=-0.10432\), product \(=(-0.6)\times(-0.10432) = 0.062592\), \((x - \bar{x})^2=(-0.6)^2 = 0.36\)
• For \((8, 1.8865)\): \(x - \bar{x}=8 - 5.6 = 2.4\), \(\log y-\bar{\log y}=1.8865 - 1.46602 = 0.42048\), product \(=2.4\times0.42048 = 1.009152\), \((x - \bar{x})^2=(2.4)^2 = 5.76\)
• For \((9, 2.0607)\): \(x - \bar{x}=9 - 5.6 = 3.4\), \(\log y-\bar{\log y}=2.0607 - 1.46602 = 0.59468\), product \(=3.4\times0.59468 = 2.021912\), \((x - \bar{x})^2=(3.4)^2 = 11.56\)

Now, sum the products: \(2.235312+0.463872 + 0.062592+1.009152+2.021912=5.79284\)

Sum the \((x - \bar{x})^2\) values: \(12.96+2.56 + 0.36+5.76+11.56 = 33.2\)

Then, \(m=\frac{5.79284}{33.2}\approx0.1745\)

Step 4: Calculate the y-intercept \(b\)

Using the formula \(b=\bar{\log y}-m\bar{x}\)
\[
b = 1.46602-0.1745\times5.6
\]
\[
b=1.46602 - 0.9772
\]
\[
b\approx0.4888
\]

Step 5: Write the linear regression equation

The linear regression equation is \(\log y=mx + b\), so \(\log y=0.1745x + 0.4888\) (values are approximate)

Step 1: Identify the data points

The data points are \((1, 65.0)\), \((3, 32.5)\), \((4, 16.3)\), \((5, 8.2)\), \((7, 1.0)\)

Step 2: Plot on a coordinate system

For the first graph (assuming a regular Cartesian plane) and the second graph (with grid lines), mark the points with \(x\) as the horizontal axis and \(y\) as the vertical axis. For example, for \((1, 65.0)\), move 1 unit on the \(x\)-axis and 65 units on the \(y\)-axis (adjusting the scale if needed), and similarly for other points.

Answer:

\(x\)	\(2\)	\(4\)	\(5\)	\(8\)	\(9\)
\(\log y\)	\(0.8451\)	\(1.1761\)	\(1.3617\)	\(1.8865\)	\(2.0607\)

Part 4e: Find a linear regression equation for \(\boldsymbol{(x, \log y)}\)