Question

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

x | y
10 | 9.15
8 | 8.15
13 | 8.74
9 | 8.78
11 | 9.27
14 | 8.09
6 | 6.12
4 | 3.09
12 | 9.13
7 | 7.27
5 | 4.74

a. construct a scatterplot. choose the correct graph below

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 =
(round to three decimal places as needed)

(there are also some tables and graphs related to critical values and scatterplot options, but the main text is as above)

Explanation:

Step1: List the data points

We have the following pairs of \( (x, y) \) values: \( (10, 9.15), (8, 8.15), (13, 8.74), (9, 8.78), (11, 9.27), (14, 8.09), (6, 6.12), (4, 3.09), (12, 9.13), (7, 7.27), (5, 4.74) \). Wait, let's count the number of points. Let's list them properly:

\( x: 10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5 \)

\( y: 9.15, 8.15, 8.74, 8.78, 9.27, 8.09, 6.12, 3.09, 9.13, 7.27, 4.74 \)

So \( n = 11 \) (number of data points).

Step2: Calculate necessary sums

First, calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), \( \sum y^2 \).

- \( \sum x = 10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5 \)
\( = (10+8)+(13+9)+(11+14)+(6+4)+(12+7)+(5) \)
\( = 18 + 22 + 25 + 10 + 19 + 5 = 18+22=40; 40+25=65; 65+10=75; 75+19=94; 94+5=99 \). So \( \sum x = 99 \).

- \( \sum y = 9.15 + 8.15 + 8.74 + 8.78 + 9.27 + 8.09 + 6.12 + 3.09 + 9.13 + 7.27 + 4.74 \)
Let's add step by step:
\( 9.15 + 8.15 = 17.3 \)
\( 17.3 + 8.74 = 26.04 \)
\( 26.04 + 8.78 = 34.82 \)
\( 34.82 + 9.27 = 44.09 \)
\( 44.09 + 8.09 = 52.18 \)
\( 52.18 + 6.12 = 58.3 \)
\( 58.3 + 3.09 = 61.39 \)
\( 61.39 + 9.13 = 70.52 \)
\( 70.52 + 7.27 = 77.79 \)
\( 77.79 + 4.74 = 82.53 \). So \( \sum y = 82.53 \).

- \( \sum xy \): Multiply each \( x \) and \( y \) and sum.
\( 10*9.15 = 91.5 \)
\( 8*8.15 = 65.2 \)
\( 13*8.74 = 113.62 \)
\( 9*8.78 = 79.02 \)
\( 11*9.27 = 101.97 \)
\( 14*8.09 = 113.26 \)
\( 6*6.12 = 36.72 \)
\( 4*3.09 = 12.36 \)
\( 12*9.13 = 109.56 \)
\( 7*7.27 = 50.89 \)
\( 5*4.74 = 23.7 \)
Now sum these: \( 91.5 + 65.2 = 156.7; +113.62 = 270.32; +79.02 = 349.34; +101.97 = 451.31; +113.26 = 564.57; +36.72 = 601.29; +12.36 = 613.65; +109.56 = 723.21; +50.89 = 774.1; +23.7 = 797.8 \). So \( \sum xy = 797.8 \).

- \( \sum x^2 \): Square each \( x \) and sum.
\( 10^2 = 100 \)
\( 8^2 = 64 \)
\( 13^2 = 169 \)
\( 9^2 = 81 \)
\( 11^2 = 121 \)
\( 14^2 = 196 \)
\( 6^2 = 36 \)
\( 4^2 = 16 \)
\( 12^2 = 144 \)
\( 7^2 = 49 \)
\( 5^2 = 25 \)
Sum: \( 100 + 64 = 164; +169 = 333; +81 = 414; +121 = 535; +196 = 731; +36 = 767; +16 = 783; +144 = 927; +49 = 976; +25 = 1001 \). So \( \sum x^2 = 1001 \).

- \( \sum y^2 \): Square each \( y \) and sum.
\( 9.15^2 = 83.7225 \)
\( 8.15^2 = 66.4225 \)
\( 8.74^2 = 76.3876 \)
\( 8.78^2 = 77.0884 \)
\( 9.27^2 = 85.9329 \)
\( 8.09^2 = 65.4481 \)
\( 6.12^2 = 37.4544 \)
\( 3.09^2 = 9.5481 \)
\( 9.13^2 = 83.3569 \)
\( 7.27^2 = 52.8529 \)
\( 4.74^2 = 22.4676 \)
Sum these: \( 83.7225 + 66.4225 = 150.145; +76.3876 = 226.5326; +77.0884 = 303.621; +85.9329 = 389.5539; +65.4481 = 455.002; +37.4544 = 492.4564; +9.5481 = 502.0045; +83.3569 = 585.3614; +52.8529 = 638.2143; +22.4676 = 660.6819 \). So \( \sum y^2 = 660.6819 \).

Step3: Use the correlation coefficient formula

The formula for the linear correlation coefficient \( r \) is:

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

Plug in the values:

- Numerator: \( n\sum xy - (\sum x)(\sum y) = 11*797.8 - 99*82.53 \)
Calculate \( 11*797.8 = 8775.8 \)
Calculate \( 99*82.53 = (100 - 1)*82.53 = 8253 - 82.53 = 8170.47 \)
Numerator: \( 8775.8 - 8170.47 = 605.33 \)

- Denominator: First part inside sqrt: \( n\sum x^2 - (\sum x)^2 = 11*1001 - 99^2 \)
\( 11*1001 = 11011 \)
\( 99^2 = 9801 \)
First part: \( 11011 - 9801 = 1210 \)

Second part inside sqrt: \( n\sum y^2 - (\sum y)^2 = 11*660.6819 - (82.53)^2 \)
Calculate \( 11*660.6819 = 7267.5009 \)
\( 82.53^2 = 6811.2009 \)
Second part: \( 7267.5009 - 6811.2009 = 456.3 \)

Now denominator: \( \sqrt{1210 * 456.3} \)
Calculate \( 1210 * 456.3 = 1210 * 456 + 1210 * 0.3 = 551760 + 363 = 552123 \)
Then \( \sqrt{552123} \approx 743.05 \) (let's check: \( 743^2 = 743*743 = (700 + 43)^2 = 700^2 + 2*700*43 + 43^2 = 490000 + 60200 + 1849 = 552049 \). Close to 552123. The difference is 552123 - 552049 = 74. So \( \sqrt{552123} \approx 743.05 + (74)/(2*743.05) \approx 743.05 + 0.05 = 743.1 \) (approximate).

Step4: Calculate \( r \)

Now, \( r = \frac{605.33}{\sqrt{1210 * 456.3}} \approx \frac{605.33}{743.1} \approx 0.8146 \). Wait, but let's check the calculations again because maybe I made a mistake in sums.

Wait, let's recheck the sum of \( x \): 10,8,13,9,11,14,6,4,12,7,5. Let's add again:

10 + 8 = 18; +13 = 31; +9 = 40; +11 = 51; +14 = 65; +6 = 71; +4 = 75; +12 = 87; +7 = 94; +5 = 99. Correct.

Sum of \( y \): 9.15,8.15,8.74,8.78,9.27,8.09,6.12,3.09,9.13,7.27,4.74. Let's add again:

9.15 + 8.15 = 17.3; +8.74 = 26.04; +8.78 = 34.82; +9.27 = 44.09; +8.09 = 52.18; +6.12 = 58.3; +3.09 = 61.39; +9.13 = 70.52; +7.27 = 77.79; +4.74 = 82.53. Correct.

Sum of \( xy \): Let's recheck one by one:

10*9.15=91.5

8*8.15=65.2 (total 156.7)

13*8.74=113.62 (total 270.32)

9*8.78=79.02 (total 349.34)

11*9.27=101.97 (total 451.31)

14*8.09=113.26 (total 564.57)

6*6.12=36.72 (total 601.29)

4*3.09=12.36 (total 613.65)

12*9.13=109.56 (total 723.21)

7*7.27=50.89 (total 774.1)

5*4.74=23.7 (total 797.8). Correct.

Sum of \( x^2 \): 100,64,169,81,121,196,36,16,144,49,25. Sum: 100+64=164; +169=333; +81=414; +121=535; +196=731; +36=767; +16=783; +144=927; +49=976; +25=1001. Correct.

Sum of \( y^2 \): Let's recheck 9.15²=83.7225; 8.15²=66.4225 (sum 150.145); 8.74²=76.3876 (sum 226.5326); 8.78²=77.0884 (sum 303.621); 9.27²=85.9329 (sum 389.5539); 8.09²=65.4481 (sum 455.002); 6.12²=37.4544 (sum 492.4564); 3.09²=9.5481 (sum 502.0045); 9.13²=83.3569 (sum 585.3614); 7.27²=52.8529 (sum 638.2143); 4.74²=22.4676 (sum 660.6819). Correct.

Now, numerator: 11*797.8 - 99*82.53 = 8775.8 - (99*82.53). 99*82.53: 82.53*100=8253, minus 82.53=8170.47. So 8775.8 - 8170.47=605.33. Correct.

Denominator: sqrt( (11*1001 - 99²)(11*660.6819 - 82.53²) )

First part: 11*1001=11011; 99²=9801; 11011-9801=1210. Correct.

Second part: 11*660.6

Answer:

Step1: List the data points

We have the following pairs of \( (x, y) \) values: \( (10, 9.15), (8, 8.15), (13, 8.74), (9, 8.78), (11, 9.27), (14, 8.09), (6, 6.12), (4, 3.09), (12, 9.13), (7, 7.27), (5, 4.74) \). Wait, let's count the number of points. Let's list them properly:

\( x: 10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5 \)

\( y: 9.15, 8.15, 8.74, 8.78, 9.27, 8.09, 6.12, 3.09, 9.13, 7.27, 4.74 \)

So \( n = 11 \) (number of data points).

Step2: Calculate necessary sums

First, calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), \( \sum y^2 \).

• \( \sum x = 10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5 \)

\( = (10+8)+(13+9)+(11+14)+(6+4)+(12+7)+(5) \)
\( = 18 + 22 + 25 + 10 + 19 + 5 = 18+22=40; 40+25=65; 65+10=75; 75+19=94; 94+5=99 \). So \( \sum x = 99 \).

• \( \sum y = 9.15 + 8.15 + 8.74 + 8.78 + 9.27 + 8.09 + 6.12 + 3.09 + 9.13 + 7.27 + 4.74 \)

Let's add step by step:
\( 9.15 + 8.15 = 17.3 \)
\( 17.3 + 8.74 = 26.04 \)
\( 26.04 + 8.78 = 34.82 \)
\( 34.82 + 9.27 = 44.09 \)
\( 44.09 + 8.09 = 52.18 \)
\( 52.18 + 6.12 = 58.3 \)
\( 58.3 + 3.09 = 61.39 \)
\( 61.39 + 9.13 = 70.52 \)
\( 70.52 + 7.27 = 77.79 \)
\( 77.79 + 4.74 = 82.53 \). So \( \sum y = 82.53 \).

• \( \sum xy \): Multiply each \( x \) and \( y \) and sum.

\( 10*9.15 = 91.5 \)
\( 8*8.15 = 65.2 \)
\( 13*8.74 = 113.62 \)
\( 9*8.78 = 79.02 \)
\( 11*9.27 = 101.97 \)
\( 14*8.09 = 113.26 \)
\( 6*6.12 = 36.72 \)
\( 4*3.09 = 12.36 \)
\( 12*9.13 = 109.56 \)
\( 7*7.27 = 50.89 \)
\( 5*4.74 = 23.7 \)
Now sum these: \( 91.5 + 65.2 = 156.7; +113.62 = 270.32; +79.02 = 349.34; +101.97 = 451.31; +113.26 = 564.57; +36.72 = 601.29; +12.36 = 613.65; +109.56 = 723.21; +50.89 = 774.1; +23.7 = 797.8 \). So \( \sum xy = 797.8 \).

• \( \sum x^2 \): Square each \( x \) and sum.

\( 10^2 = 100 \)
\( 8^2 = 64 \)
\( 13^2 = 169 \)
\( 9^2 = 81 \)
\( 11^2 = 121 \)
\( 14^2 = 196 \)
\( 6^2 = 36 \)
\( 4^2 = 16 \)
\( 12^2 = 144 \)
\( 7^2 = 49 \)
\( 5^2 = 25 \)
Sum: \( 100 + 64 = 164; +169 = 333; +81 = 414; +121 = 535; +196 = 731; +36 = 767; +16 = 783; +144 = 927; +49 = 976; +25 = 1001 \). So \( \sum x^2 = 1001 \).

• \( \sum y^2 \): Square each \( y \) and sum.

\( 9.15^2 = 83.7225 \)
\( 8.15^2 = 66.4225 \)
\( 8.74^2 = 76.3876 \)
\( 8.78^2 = 77.0884 \)
\( 9.27^2 = 85.9329 \)
\( 8.09^2 = 65.4481 \)
\( 6.12^2 = 37.4544 \)
\( 3.09^2 = 9.5481 \)
\( 9.13^2 = 83.3569 \)
\( 7.27^2 = 52.8529 \)
\( 4.74^2 = 22.4676 \)
Sum these: \( 83.7225 + 66.4225 = 150.145; +76.3876 = 226.5326; +77.0884 = 303.621; +85.9329 = 389.5539; +65.4481 = 455.002; +37.4544 = 492.4564; +9.5481 = 502.0045; +83.3569 = 585.3614; +52.8529 = 638.2143; +22.4676 = 660.6819 \). So \( \sum y^2 = 660.6819 \).

Step3: Use the correlation coefficient formula

The formula for the linear correlation coefficient \( r \) is:

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

Plug in the values:

• Numerator: \( n\sum xy - (\sum x)(\sum y) = 11797.8 - 9982.53 \)

Calculate \( 11*797.8 = 8775.8 \)
Calculate \( 9982.53 = (100 - 1)82.53 = 8253 - 82.53 = 8170.47 \)
Numerator: \( 8775.8 - 8170.47 = 605.33 \)

• Denominator: First part inside sqrt: \( n\sum x^2 - (\sum x)^2 = 11*1001 - 99^2 \)

\( 11*1001 = 11011 \)
\( 99^2 = 9801 \)
First part: \( 11011 - 9801 = 1210 \)

Second part inside sqrt: \( n\sum y^2 - (\sum y)^2 = 11*660.6819 - (82.53)^2 \)
Calculate \( 11*660.6819 = 7267.5009 \)
\( 82.53^2 = 6811.2009 \)
Second part: \( 7267.5009 - 6811.2009 = 456.3 \)

Now denominator: \( \sqrt{1210 * 456.3} \)
Calculate \( 1210 * 456.3 = 1210 * 456 + 1210 * 0.3 = 551760 + 363 = 552123 \)
Then \( \sqrt{552123} \approx 743.05 \) (let's check: \( 743^2 = 743743 = (700 + 43)^2 = 700^2 + 270043 + 43^2 = 490000 + 60200 + 1849 = 552049 \). Close to 552123. The difference is 552123 - 552049 = 74. So \( \sqrt{552123} \approx 743.05 + (74)/(2743.05) \approx 743.05 + 0.05 = 743.1 \) (approximate).

Step4: Calculate \( r \)

Now, \( r = \frac{605.33}{\sqrt{1210 * 456.3}} \approx \frac{605.33}{743.1} \approx 0.8146 \). Wait, but let's check the calculations again because maybe I made a mistake in sums.

Wait, let's recheck the sum of \( x \): 10,8,13,9,11,14,6,4,12,7,5. Let's add again:

10 + 8 = 18; +13 = 31; +9 = 40; +11 = 51; +14 = 65; +6 = 71; +4 = 75; +12 = 87; +7 = 94; +5 = 99. Correct.

Sum of \( y \): 9.15,8.15,8.74,8.78,9.27,8.09,6.12,3.09,9.13,7.27,4.74. Let's add again:

9.15 + 8.15 = 17.3; +8.74 = 26.04; +8.78 = 34.82; +9.27 = 44.09; +8.09 = 52.18; +6.12 = 58.3; +3.09 = 61.39; +9.13 = 70.52; +7.27 = 77.79; +4.74 = 82.53. Correct.

Sum of \( xy \): Let's recheck one by one:

10*9.15=91.5

8*8.15=65.2 (total 156.7)

13*8.74=113.62 (total 270.32)

9*8.78=79.02 (total 349.34)

11*9.27=101.97 (total 451.31)

14*8.09=113.26 (total 564.57)

6*6.12=36.72 (total 601.29)

4*3.09=12.36 (total 613.65)

12*9.13=109.56 (total 723.21)

7*7.27=50.89 (total 774.1)

5*4.74=23.7 (total 797.8). Correct.

Sum of \( x^2 \): 100,64,169,81,121,196,36,16,144,49,25. Sum: 100+64=164; +169=333; +81=414; +121=535; +196=731; +36=767; +16=783; +144=927; +49=976; +25=1001. Correct.

Sum of \( y^2 \): Let's recheck 9.15²=83.7225; 8.15²=66.4225 (sum 150.145); 8.74²=76.3876 (sum 226.5326); 8.78²=77.0884 (sum 303.621); 9.27²=85.9329 (sum 389.5539); 8.09²=65.4481 (sum 455.002); 6.12²=37.4544 (sum 492.4564); 3.09²=9.5481 (sum 502.0045); 9.13²=83.3569 (sum 585.3614); 7.27²=52.8529 (sum 638.2143); 4.74²=22.4676 (sum 660.6819). Correct.

Now, numerator: 11797.8 - 9982.53 = 8775.8 - (9982.53). 9982.53: 82.53*100=8253, minus 82.53=8170.47. So 8775.8 - 8170.47=605.33. Correct.

Denominator: sqrt( (111001 - 99²)(11660.6819 - 82.53²) )

First part: 11*1001=11011; 99²=9801; 11011-9801=1210. Correct.

Second part: 11*660.6