Question

listed below are annual data for various years. the data are the numbers of points scored in the super bowl. construct a scatterplot, find the value of the p - value using α = 0.05. is there sufficient evidence to conclude that the two variables? would it be reasonable to expect a correlation?
car sales: 8179, 8218, 8518, 8992, 8636, 8535, 8275
super bowl points: 52, 70, 43, 74, 43, 57, 56
the linear correlation coefficient is r =
(round to three decimal places as needed.)

Explanation:

Step1: Identify Variables

Let \( x \) be Car Sales (values: 8179, 8218, 8518, 8992, 8636, 8535, 8275) and \( y \) be Super Bowl Points (values: 52, 70, 43, 74, 43, 57, 56).

Step2: Calculate Necessary Sums

- \( \sum x = 8179 + 8218 + 8518 + 8992 + 8636 + 8535 + 8275 = 59353 \)
- \( \sum y = 52 + 70 + 43 + 74 + 43 + 57 + 56 = 395 \)
- \( \sum xy = (8179×52)+(8218×70)+(8518×43)+(8992×74)+(8636×43)+(8535×57)+(8275×56) \)
\( = 425308 + 575260 + 366274 + 665408 + 371348 + 486495 + 463400 = 3353493 \)
- \( \sum x^2 = 8179^2 + 8218^2 + 8518^2 + 8992^2 + 8636^2 + 8535^2 + 8275^2 \)
\( = 66896041 + 67535524 + 72556324 + 80856064 + 74580496 + 72846225 + 68475625 = 491746299 \)
- \( \sum y^2 = 52^2 + 70^2 + 43^2 + 74^2 + 43^2 + 57^2 + 56^2 \)
\( = 2704 + 4900 + 1849 + 5476 + 1849 + 3249 + 3136 = 23163 \)

Step3: Apply Correlation Formula

The formula for the linear correlation coefficient \( r \) is:
\[ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
where \( n = 7 \).

- Numerator: \( 7×3353493 - 59353×395 = 23474451 - 23444435 = 30016 \)
- Denominator part 1: \( 7×491746299 - 59353^2 = 3442224093 - 3522788609 = -80564516 \) (Wait, no, recalculate \( 59353^2 \): \( 59353×59353 = 3522788609 \), \( 7×491746299 = 3442224093 \). Wait, that can't be. Wait, I must have miscalculated \( \sum x \). Wait, 8179 + 8218 = 16397; +8518=24915; +8992=33907; +8636=42543; +8535=51078; +8275=59353. Correct. Wait, \( 7×491746299 = 3442224093 \), \( 3442224093 - 3522788609 = -80564516 \). Denominator part 2: \( 7×23163 - 395^2 = 162141 - 156025 = 6116 \)
- Denominator: \( \sqrt{|-80564516|×6116} = \sqrt{80564516×6116} \). Wait, no, I must have messed up \( \sum x^2 \). Let's recalculate \( \sum x^2 \):
- \( 8179^2 = 8179×8179 = 66896041 \)
- \( 8218^2 = 67535524 \)
- \( 8518^2 = 72556324 \)
- \( 8992^2 = 80856064 \)
- \( 8636^2 = 74580496 \)
- \( 8535^2 = 72846225 \)
- \( 8275^2 = 68475625 \)
Sum: 66896041 + 67535524 = 134431565; +72556324=206987889; +80856064=287843953; +74580496=362424449; +72846225=435270674; +68475625=503746299. Ah! I had a typo earlier, \( \sum x^2 = 503746299 \), not 491746299. So \( 7×503746299 = 3526224093 \)
Now denominator part 1: \( 3526224093 - 59353^2 = 3526224093 - 3522788609 = 3435484 \)
Denominator part 2: \( 7×23163 - 395^2 = 162141 - 156025 = 6116 \)
Denominator: \( \sqrt{3435484×6116} = \sqrt{3435484×6116} \). Calculate \( 3435484×6116 \): Let's factor 3435484 = 4×858871, 6116=4×1529. So \( \sqrt{16×858871×1529} = 4\sqrt{858871×1529} \). But better to use calculator steps. Alternatively, use the formula correctly.

Wait, let's use a better approach. Let's compute \( \bar{x} = \frac{59353}{7} ≈ 8479 \), \( \bar{y} = \frac{395}{7} ≈ 56.4286 \)

\( \sum (x - \bar{x})(y - \bar{y}) = \sum xy - n\bar{x}\bar{y} = 3353493 - 7×8479×56.4286 \)

Calculate \( 7×8479 = 59353 \), \( 59353×56.4286 ≈ 59353×56 + 59353×0.4286 ≈ 3323768 + 25440 ≈ 3349208 \)

So \( \sum (x - \bar{x})(y - \bar{y}) ≈ 3353493 - 3349208 = 4285 \)

\( \sum (x - \bar{x})^2 = \sum x^2 - n\bar{x}^2 = 503746299 - 7×(8479)^2 \)

\( 8479^2 = 71893441 \), \( 7×71893441 = 503254087 \)

\( \sum (x - \bar{x})^2 = 503746299 - 503254087 = 492212 \)

\( \sum (y - \bar{y})^2 = \sum y^2 - n\bar{y}^2 = 23163 - 7×(56.4286)^2 \)

\( 56.4286^2 ≈ 3184.14 \), \( 7×3184.14 ≈ 22288.98 \)

\( \sum (y - \bar{y})^2 ≈ 23163 - 22288.98 = 874.02 \)

Now, \( r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} = \frac{4285}{\sqrt{492212×874.02}} \)

Calculate denominator: \( \sqrt{492212×874.02} ≈ \sqrt{430,200,000} ≈ 20741.26 \)

Then \( r ≈ \frac{4285}{20741.26} ≈ 0.207 \)

Wait, but let's recalculate with precise numbers.

Wait, \( \sum xy = 8179×52 = 425308 \); 8218×70=575260; 8518×43=366274; 8992×74=665408; 8636×43=371348; 8535×57=486495; 8275×56=463400. Sum: 425308+575260=1,000,568; +366,274=1,366,842; +665,408=2,032,250; +371,348=2,403,598; +486,495=2,890,093; +463,400=3,353,493. Correct.

\( n=7 \), \( \sum x=59353 \), \( \sum y=395 \), \( \sum x^2=503746299 \), \( \sum y^2=23163 \)

So:

Numerator: \( 7×3353493 - 59353×395 = 23474451 - 23444435 = 30016 \) (Wait, earlier mistake in \( \bar{x}\bar{y} \) calculation. \( 59353×395 = 59353×(400 - 5) = 59353×400 - 59353×5 = 23,741,200 - 296,765 = 23,444,435 \). \( 7×3353493 = 23,474,451 \). So numerator is 23,474,451 - 23,444,435 = 30,016. Correct.

Denominator part 1: \( 7×503746299 - 59353^2 = 3,526,224,093 - 3,522,788,609 = 3,435,484 \) (since \( 59353^2 = (59000 + 353)^2 = 59000^2 + 2×59000×353 + 353^2 = 3,481,000,000 + 41,054,000 + 124,609 = 3,522,178,609 \)? Wait, no, 59353×59353: let's use calculator-like steps. 59353×59353:

= (60000 - 647)² = 60000² - 2×60000×647 + 647² = 3,600,000,000 - 77,640,000 + 418,609 = 3,522,778,609. Ah! I had a typo earlier, 59353² is 3,522,778,609, not 3,522,788,609. So \( 7×503,746,299 = 3,526,224,093 \). Then \( 3,526,224,093 - 3,522,778,609 = 3,445,484 \).

Denominator part 2: \( 7×23,163 - 395² = 162,141 - 156,025 = 6,116 \).

Now denominator: \( \sqrt{3,445,484×6,116} \). Let's compute 3,445,484×6,116:

3,445,484×6,000 = 20,672,904,000

3,445,484×116 = 3,445,484×(100+16) = 344,548,400 + 55,127,744 = 399,676,144

Total: 20,672,904,000 + 399,676,144 = 21,072,580,144

Square root of 21,072,580,144: Let's see, 145,164² = (145,000 + 164)² = 145,000² + 2×145,000×164 + 164² = 21,025,000,000 + 47,840,000 + 26,896 = 21,072,866,896. Close, but our product is 21,072,580,144, which is less. Let's try 145,150² = (145,000 + 150)² = 21,025,000,000 + 43,500,000 + 22,500 = 21,068,522,500. Still less. 145,160² = 145,150² + 2×145,150×10 + 10² = 21,068,522,500 + 2,903,000 + 100 = 21,071,425,600. 145,163²

Answer:

Step1: Identify Variables

Let \( x \) be Car Sales (values: 8179, 8218, 8518, 8992, 8636, 8535, 8275) and \( y \) be Super Bowl Points (values: 52, 70, 43, 74, 43, 57, 56).

Step2: Calculate Necessary Sums

• \( \sum x = 8179 + 8218 + 8518 + 8992 + 8636 + 8535 + 8275 = 59353 \)
• \( \sum y = 52 + 70 + 43 + 74 + 43 + 57 + 56 = 395 \)
• \( \sum xy = (8179×52)+(8218×70)+(8518×43)+(8992×74)+(8636×43)+(8535×57)+(8275×56) \)

\( = 425308 + 575260 + 366274 + 665408 + 371348 + 486495 + 463400 = 3353493 \)

• \( \sum x^2 = 8179^2 + 8218^2 + 8518^2 + 8992^2 + 8636^2 + 8535^2 + 8275^2 \)

\( = 66896041 + 67535524 + 72556324 + 80856064 + 74580496 + 72846225 + 68475625 = 491746299 \)

• \( \sum y^2 = 52^2 + 70^2 + 43^2 + 74^2 + 43^2 + 57^2 + 56^2 \)

\( = 2704 + 4900 + 1849 + 5476 + 1849 + 3249 + 3136 = 23163 \)

Step3: Apply Correlation Formula

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

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

where \( n = 7 \).

• Numerator: \( 7×3353493 - 59353×395 = 23474451 - 23444435 = 30016 \)
• Denominator part 1: \( 7×491746299 - 59353^2 = 3442224093 - 3522788609 = -80564516 \) (Wait, no, recalculate \( 59353^2 \): \( 59353×59353 = 3522788609 \), \( 7×491746299 = 3442224093 \). Wait, that can't be. Wait, I must have miscalculated \( \sum x \). Wait, 8179 + 8218 = 16397; +8518=24915; +8992=33907; +8636=42543; +8535=51078; +8275=59353. Correct. Wait, \( 7×491746299 = 3442224093 \), \( 3442224093 - 3522788609 = -80564516 \). Denominator part 2: \( 7×23163 - 395^2 = 162141 - 156025 = 6116 \)
• Denominator: \( \sqrt{|-80564516|×6116} = \sqrt{80564516×6116} \). Wait, no, I must have messed up \( \sum x^2 \). Let's recalculate \( \sum x^2 \):
• \( 8179^2 = 8179×8179 = 66896041 \)
• \( 8218^2 = 67535524 \)
• \( 8518^2 = 72556324 \)
• \( 8992^2 = 80856064 \)
• \( 8636^2 = 74580496 \)
• \( 8535^2 = 72846225 \)
• \( 8275^2 = 68475625 \)

Sum: 66896041 + 67535524 = 134431565; +72556324=206987889; +80856064=287843953; +74580496=362424449; +72846225=435270674; +68475625=503746299. Ah! I had a typo earlier, \( \sum x^2 = 503746299 \), not 491746299. So \( 7×503746299 = 3526224093 \)
Now denominator part 1: \( 3526224093 - 59353^2 = 3526224093 - 3522788609 = 3435484 \)
Denominator part 2: \( 7×23163 - 395^2 = 162141 - 156025 = 6116 \)
Denominator: \( \sqrt{3435484×6116} = \sqrt{3435484×6116} \). Calculate \( 3435484×6116 \): Let's factor 3435484 = 4×858871, 6116=4×1529. So \( \sqrt{16×858871×1529} = 4\sqrt{858871×1529} \). But better to use calculator steps. Alternatively, use the formula correctly.

Wait, let's use a better approach. Let's compute \( \bar{x} = \frac{59353}{7} ≈ 8479 \), \( \bar{y} = \frac{395}{7} ≈ 56.4286 \)

\( \sum (x - \bar{x})(y - \bar{y}) = \sum xy - n\bar{x}\bar{y} = 3353493 - 7×8479×56.4286 \)

Calculate \( 7×8479 = 59353 \), \( 59353×56.4286 ≈ 59353×56 + 59353×0.4286 ≈ 3323768 + 25440 ≈ 3349208 \)

So \( \sum (x - \bar{x})(y - \bar{y}) ≈ 3353493 - 3349208 = 4285 \)

\( \sum (x - \bar{x})^2 = \sum x^2 - n\bar{x}^2 = 503746299 - 7×(8479)^2 \)

\( 8479^2 = 71893441 \), \( 7×71893441 = 503254087 \)

\( \sum (x - \bar{x})^2 = 503746299 - 503254087 = 492212 \)

\( \sum (y - \bar{y})^2 = \sum y^2 - n\bar{y}^2 = 23163 - 7×(56.4286)^2 \)

\( 56.4286^2 ≈ 3184.14 \), \( 7×3184.14 ≈ 22288.98 \)

\( \sum (y - \bar{y})^2 ≈ 23163 - 22288.98 = 874.02 \)

Now, \( r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} = \frac{4285}{\sqrt{492212×874.02}} \)

Calculate denominator: \( \sqrt{492212×874.02} ≈ \sqrt{430,200,000} ≈ 20741.26 \)

Then \( r ≈ \frac{4285}{20741.26} ≈ 0.207 \)

Wait, but let's recalculate with precise numbers.

Wait, \( \sum xy = 8179×52 = 425308 \); 8218×70=575260; 8518×43=366274; 8992×74=665408; 8636×43=371348; 8535×57=486495; 8275×56=463400. Sum: 425308+575260=1,000,568; +366,274=1,366,842; +665,408=2,032,250; +371,348=2,403,598; +486,495=2,890,093; +463,400=3,353,493. Correct.

\( n=7 \), \( \sum x=59353 \), \( \sum y=395 \), \( \sum x^2=503746299 \), \( \sum y^2=23163 \)

So:

Numerator: \( 7×3353493 - 59353×395 = 23474451 - 23444435 = 30016 \) (Wait, earlier mistake in \( \bar{x}\bar{y} \) calculation. \( 59353×395 = 59353×(400 - 5) = 59353×400 - 59353×5 = 23,741,200 - 296,765 = 23,444,435 \). \( 7×3353493 = 23,474,451 \). So numerator is 23,474,451 - 23,444,435 = 30,016. Correct.

Denominator part 1: \( 7×503746299 - 59353^2 = 3,526,224,093 - 3,522,788,609 = 3,435,484 \) (since \( 59353^2 = (59000 + 353)^2 = 59000^2 + 2×59000×353 + 353^2 = 3,481,000,000 + 41,054,000 + 124,609 = 3,522,178,609 \)? Wait, no, 59353×59353: let's use calculator-like steps. 59353×59353:

= (60000 - 647)² = 60000² - 2×60000×647 + 647² = 3,600,000,000 - 77,640,000 + 418,609 = 3,522,778,609. Ah! I had a typo earlier, 59353² is 3,522,778,609, not 3,522,788,609. So \( 7×503,746,299 = 3,526,224,093 \). Then \( 3,526,224,093 - 3,522,778,609 = 3,445,484 \).

Denominator part 2: \( 7×23,163 - 395² = 162,141 - 156,025 = 6,116 \).

Now denominator: \( \sqrt{3,445,484×6,116} \). Let's compute 3,445,484×6,116:

3,445,484×6,000 = 20,672,904,000

3,445,484×116 = 3,445,484×(100+16) = 344,548,400 + 55,127,744 = 399,676,144

Total: 20,672,904,000 + 399,676,144 = 21,072,580,144

Square root of 21,072,580,144: Let's see, 145,164² = (145,000 + 164)² = 145,000² + 2×145,000×164 + 164² = 21,025,000,000 + 47,840,000 + 26,896 = 21,072,866,896. Close, but our product is 21,072,580,144, which is less. Let's try 145,150² = (145,000 + 150)² = 21,025,000,000 + 43,500,000 + 22,500 = 21,068,522,500. Still less. 145,160² = 145,150² + 2×145,150×10 + 10² = 21,068,522,500 + 2,903,000 + 100 = 21,071,425,600. 145,163²