Question

use the given data set to complete parts (a) through (c) below. (use \\(\alpha = 0.05\\).)
\\(\

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \\hline x & 10 & 8 & 13 & 9 & 11 & 14 & 6 & 4 & 12 & 7 & 5 \\\\ \\hline y & 9.14 & 8.14 & 8.74 & 8.76 & 9.26 & 8.11 & 6.14 & 3.09 & 9.13 & 7.27 & 4.74 \\\\ \\hline \\end{array}$$

\\)

a. construct a scatterplot. choose the correct graph below.
\\(\bigcirc\\) a. \\(\quad\bigcirc\\) b. \\(\quad\bigcirc\\) c. \\(\quad\bigcirc\\) d.
(graphs of scatterplots for a, b, c, d are shown with x and y axes from 0 to 16 and 0 to 10 respectively, with different point distributions.)

b. find the linear correlation coefficient, \\(r\\), then determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.
the linear correlation coefficient is \\(r = \square\\).
(round to three decimal places as needed.)
using the linear correlation coefficient found in the previous step, determine whether there is...

Explanation:

Step 1: Organize the data

We have the following data pairs \((x, y)\):
\((10, 9.14)\), \((8, 8.14)\), \((13, 8.74)\), \((9, 8.76)\), \((11, 9.26)\), \((14, 8.11)\), \((6, 6.14)\), \((4, 3.09)\), \((12, 9.13)\), \((7, 7.27)\), \((5, 4.74)\)

First, we calculate the necessary sums:
- \(n = 11\) (number of data points)
- \(\sum x = 10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5 = 99\)
- \(\sum y = 9.14 + 8.14 + 8.74 + 8.76 + 9.26 + 8.11 + 6.14 + 3.09 + 9.13 + 7.27 + 4.74 = 82.52\)
- \(\sum xy = (10\times9.14)+(8\times8.14)+(13\times8.74)+(9\times8.76)+(11\times9.26)+(14\times8.11)+(6\times6.14)+(4\times3.09)+(12\times9.13)+(7\times7.27)+(5\times4.74)\)
\[ \begin{align*} \sum xy&= 91.4 + 65.12 + 113.62 + 78.84 + 101.86 + 113.54 + 36.84 + 12.36 + 109.56 + 50.89 + 23.7\\ &= 91.4+65.12 = 156.52; 156.52+113.62 = 270.14; 270.14+78.84 = 348.98; 348.98+101.86 = 450.84;\\ &450.84+113.54 = 564.38; 564.38+36.84 = 601.22; 601.22+12.36 = 613.58; 613.58+109.56 = 723.14;\\ &723.14+50.89 = 774.03; 774.03+23.7 = 797.73 \end{align*} \]
- \(\sum x^2 = 10^2 + 8^2 + 13^2 + 9^2 + 11^2 + 14^2 + 6^2 + 4^2 + 12^2 + 7^2 + 5^2\)
\[ \begin{align*} \sum x^2&= 100 + 64 + 169 + 81 + 121 + 196 + 36 + 16 + 144 + 49 + 25\\ &= 100+64 = 164; 164+169 = 333; 333+81 = 414; 414+121 = 535; 535+196 = 731;\\ &731+36 = 767; 767+16 = 783; 783+144 = 927; 927+49 = 976; 976+25 = 1001 \end{align*} \]
- \(\sum y^2 = 9.14^2 + 8.14^2 + 8.74^2 + 8.76^2 + 9.26^2 + 8.11^2 + 6.14^2 + 3.09^2 + 9.13^2 + 7.27^2 + 4.74^2\)
\[ \begin{align*} \sum y^2&= 83.5396 + 66.2596 + 76.3876 + 76.7376 + 85.7476 + 65.7721 + 37.6996 + 9.5481 + 83.3569 + 52.8529 + 22.4676\\ &= 83.5396+66.2596 = 149.7992; 149.7992+76.3876 = 226.1868; 226.1868+76.7376 = 302.9244;\\ &302.9244+85.7476 = 388.672; 388.672+65.7721 = 454.4441; 454.4441+37.6996 = 492.1437;\\ &492.1437+9.5481 = 501.6918; 501.6918+83.3569 = 585.0487; 585.0487+52.8529 = 637.9016;\\ &637.9016+22.4676 = 660.3692 \end{align*} \]

Step 2: Use the formula for the linear correlation coefficient \(r\)

The formula for \(r\) is:
\[ r=\frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]

Substitute the values we calculated:
- Numerator: \(n\sum xy - (\sum x)(\sum y)=11\times797.73 - 99\times82.52\)
\[ \begin{align*} 11\times797.73&=8775.03\\ 99\times82.52&=8169.48\\ \text{Numerator}&=8775.03 - 8169.48 = 605.55 \end{align*} \]
- Denominator part 1: \(n\sum x^2 - (\sum x)^2=11\times1001 - 99^2\)
\[ \begin{align*} 11\times1001&=11011\\ 99^2&=9801\\ \text{Denominator part 1}&=11011 - 9801 = 1210 \end{align*} \]
- Denominator part 2: \(n\sum y^2 - (\sum y)^2=11\times660.3692 - 82.52^2\)
\[ \begin{align*} 11\times660.3692&=7264.0612\\ 82.52^2&=6809.5504\\ \text{Denominator part 2}&=7264.0612 - 6809.5504 = 454.5108 \end{align*} \]
- Denominator: \(\sqrt{1210\times454.5108}=\sqrt{1210\times454.5108}\)
\[ 1210\times454.5108 = 550, 1210\times454.5108=1210\times454 + 1210\times0.5108=549340+618.068 = 549958.068 \]
\(\sqrt{549958.068}\approx741.6\) (approximate value, more accurately we can calculate as follows: \(741^2 = 549081\), \(742^2=550564\), so we can use a calculator for better precision. Let's use a calculator for the square root: \(\sqrt{1210\times454.5108}=\sqrt{549958.068}\approx741.6\) (but actually, using a calculator for precise calculation: \(1210\times454.5108 = 1210\times454.5108 = 549958.068\), \(\sqrt{549958.068}\approx741.6\) (but let's do it more accurately with the formula)

Wait, actually, let's recalculate the denominator:

\(n\sum x^2 - (\sum x)^2=11\times1001 - 99^2 = 11011 - 9801 = 1210\)

\(n\sum y^2 - (\sum y)^2=11\times660.3692 - 82.52^2 = 7264.0612 - 6809.5504 = 454.5108\)

So the product inside the square root is \(1210\times454.5108 = 1210\times454.5108\)

\(1210\times454.5108 = (1200 + 10)\times454.5108 = 1200\times454.5108+10\times454.5108 = 545412.96+4545.108 = 549958.068\)

\(\sqrt{549958.068}\approx741.6\) (but let's use a calculator for the correlation coefficient formula. Alternatively, we can use a calculator or software for a more accurate calculation.

Using a calculator for the correlation coefficient:

\(r=\frac{11\times797.73 - 99\times82.52}{\sqrt{(11\times1001 - 99^2)(11\times660.3692 - 82.52^2)}}\)

\(11\times797.73 = 8775.03\), \(99\times82.52 = 8169.48\), so numerator is \(8775.03 - 8169.48 = 605.55\)

Denominator: \(\sqrt{(1210)(454.5108)}=\sqrt{549958.068}\approx741.6\)

So \(r=\frac{605.55}{741.6}\approx0.816\) (rounded to three decimal places)

Answer:

The linear correlation coefficient is \(r \approx \boxed{0.816}\)