Question

find the correlation coefficient, r, of the data described below.
leroy runs a bakery whose menu lists the number of calories in each item. he wondered whether this information affects which pastries are purchased, so he collected some sales data to find out.
leroy kept track of how many calories each pastry has, x, and how many times that pastry had been purchased in the last week, y.
round your answer to the nearest thousandth.
r =

Explanation:

Step1: List the data points

We have the following data points for \( x \) (calories) and \( y \) (sales):
\( (121, 114) \), \( (145, 240) \), \( (181, 461) \), \( (189, 457) \), \( (210, 425) \), \( (243, 374) \)

Step2: Calculate necessary sums

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

- \( \sum x = 121 + 145 + 181 + 189 + 210 + 243 = 1089 \)
- \( \sum y = 114 + 240 + 461 + 457 + 425 + 374 = 2071 \)
- \( \sum xy = (121\times114)+(145\times240)+(181\times461)+(189\times457)+(210\times425)+(243\times374) \)
\( = 13794 + 34800 + 83441 + 86373 + 89250 + 90882 = 398540 \)
- \( \sum x^2 = 121^2 + 145^2 + 181^2 + 189^2 + 210^2 + 243^2 \)
\( = 14641 + 21025 + 32761 + 35721 + 44100 + 59049 = 207297 \)
- \( \sum y^2 = 114^2 + 240^2 + 461^2 + 457^2 + 425^2 + 374^2 \)
\( = 12996 + 57600 + 212521 + 208849 + 180625 + 140076 = 812667 \)

Step3: Use the correlation coefficient formula

The formula for the 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 \( n = 6 \) (number of data points)

First, calculate the numerator:
\( n\sum xy - \sum x \sum y = 6\times398540 - 1089\times2071 \)
\( = 2391240 - 2255319 = 135921 \)

Then, calculate the denominator:
- \( n\sum x^2 - (\sum x)^2 = 6\times207297 - 1089^2 \)
\( = 1243782 - 1185921 = 57861 \)
- \( n\sum y^2 - (\sum y)^2 = 6\times812667 - 2071^2 \)
\( = 4876002 - 4289041 = 586961 \)
- Now, the denominator is \( \sqrt{57861\times586961} \)
\( \sqrt{57861\times586961} \approx \sqrt{34060000000} \approx 184553.51 \) (approximate value, more accurately calculated as follows)
First, \( 57861\times586961 = 57861\times586961 \approx 34060000000 \) (but we calculate it precisely)
\( 57861\times586961 = 57861\times(500000 + 80000 + 6000 + 900 + 60 + 1) \)
\( = 57861\times500000 + 57861\times80000 + 57861\times6000 + 57861\times900 + 57861\times60 + 57861\times1 \)
\( = 28930500000 + 4628880000 + 347166000 + 52074900 + 3471660 + 57861 \)
\( = 28930500000 + 4628880000 = 33559380000 \)
\( 33559380000 + 347166000 = 33906546000 \)
\( 33906546000 + 52074900 = 33958620900 \)
\( 33958620900 + 3471660 = 33962092560 \)
\( 33962092560 + 57861 = 33962150421 \)
Then, \( \sqrt{33962150421} \approx 184288.23 \) (more accurate)

Now, \( r = \frac{135921}{184288.23} \approx 0.737 \)

Wait, let's recalculate the denominator more accurately:

First, \( n\sum x^2 - (\sum x)^2 = 6*207297 - 1089^2 = 1243782 - 1185921 = 57861 \)

\( n\sum y^2 - (\sum y)^2 = 6*812667 - 2071^2 = 4876002 - 4289041 = 586961 \)

Now, \( 57861*586961 = 57861*586961 \)

Let's compute \( 57861*586961 \):

First, note that \( 57861 = 3*19287 = 3*3*6429 = 3*3*3*2143 \)

\( 586961 \) is a prime? Maybe not, but we can use a calculator approach.

Alternatively, use the formula step by step with more precision.

Let's recalculate the numerator and denominator with more precision.

\( n = 6 \)

\( \sum x = 1089 \), \( \sum y = 2071 \)

\( \sum xy = 398540 \)

\( \sum x^2 = 207297 \)

\( \sum y^2 = 812667 \)

Numerator: \( 6*398540 - 1089*2071 = 2391240 - (1089*2071) \)

Calculate \( 1089*2071 \):

\( 1000*2071 = 2071000 \)

\( 80*2071 = 165680 \)

\( 9*2071 = 18639 \)

Sum: \( 2071000 + 165680 = 2236680 + 18639 = 2255319 \)

So numerator: \( 2391240 - 2255319 = 135921 \)

Denominator part 1: \( n\sum x^2 - (\sum x)^2 = 6*207297 - 1089^2 = 1243782 - 1185921 = 57861 \)

Denominator part 2: \( n\sum y^2 - (\sum y)^2 = 6*812667 - 2071^2 = 4876002 - 4289041 = 586961 \)

Now, the denominator is \( \sqrt{57861 * 586961} \)

Calculate \( 57861 * 586961 \):

\( 57861 * 586961 = 57861 * (586000 + 961) = 57861*586000 + 57861*961 \)

\( 57861*586000 = 57861*586*1000 \)

\( 57861*586 = 57861*(500 + 80 + 6) = 57861*500 + 57861*80 + 57861*6 \)

\( = 28930500 + 4628880 + 347166 = 28930500 + 4628880 = 33559380 + 347166 = 33906546 \)

So \( 57861*586000 = 33906546*1000 = 33906546000 \)

\( 57861*961 = 57861*(900 + 60 + 1) = 57861*900 + 57861*60 + 57861*1 \)

\( = 52074900 + 3471660 + 57861 = 52074900 + 3471660 = 55546560 + 57861 = 55604421 \)

So total \( 57861*586961 = 33906546000 + 55604421 = 33962150421 \)

Now, \( \sqrt{33962150421} \approx 184288.23 \) (using a calculator, \( \sqrt{33962150421} \approx 184288.23 \))

Now, \( r = \frac{135921}{184288.23} \approx 0.737 \) (rounded to the nearest thousandth)

Wait, but let's check with a calculator for more accuracy.

Alternatively, use the formula in a calculator-like approach:

The correlation coefficient formula can also be calculated using the following steps:

First, calculate the mean of \( x \), \( \bar{x} = \frac{\sum x}{n} = \frac{1089}{6} = 181.5 \)

Mean of \( y \), \( \bar{y} = \frac{\sum y}{n} = \frac{2071}{6} \approx 345.1667 \)

Then, calculate the numerator as \( \sum (x - \bar{x})(y - \bar{y}) \)

And the denominator as \( \sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2} \)

Let's calculate \( (x - \bar{x}) \) and \( (y - \bar{y}) \) for each data point:

1. \( x = 121 \), \( y = 114 \)
\( x - \bar{x} = 121 - 181.5 = -60.5 \)
\( y - \bar{y} = 114 - 345.1667 = -231.1667 \)
\( (x - \bar{x})(y - \bar{y}) = (-60.5)(-231.1667) \approx 13985.595 \)
\( (x - \bar{x})^2 = (-60.5)^2 = 3660.25 \)
\( (y - \bar{y})^2 = (-231.1667)^2 \approx 53438.0 \)

2. \( x = 145 \), \( y = 240 \)
\( x - \bar{x} = 145 - 181.5 = -36.5 \)
\( y - \bar{y} = 240 - 345.1667 = -105.1667 \)
\( (x - \bar{x})(y - \bar{y}) = (-36.5)(-105.1667) \approx 3848.585 \)
\( (x - \bar{x})^2 = (-36.5)^2 = 1332.25 \)
\( (y - \bar{y})^2 = (-105.1667)^2 \approx 11059.0 \)

3. \( x = 181 \), \( y = 461 \)
\( x - \bar{x} = 181 - 181.5 = -0.5 \)
\( y - \bar{y} = 461 - 345.1667 = 115.8333 \)
\( (x - \bar{x})(y - \bar{y}) = (-0.5)(115.8333) \approx -57.9167 \)
\( (x - \bar{x})^2 = (-0.5)^2 = 0.25 \)
\( (y - \bar{y})^2 = (115.8333)^2 \approx 13417.0 \)

4. \( x = 189 \), \( y = 457 \)
\( x - \bar{x} = 189 - 181.5 = 7.5 \)
\( y - \bar{y} = 457 - 345.1667 = 111.8333 \)
\( (x - \bar{x})(y - \bar{y}) = (7.5)(111.8333) \approx 838.75 \)
\( (x - \bar{x})^2 = (7.5)^2 = 56.25 \)
\( (y - \bar{y})^2 = (111.8333)^2 \approx 12507.0 \)

5. \( x = 210 \), \( y = 425 \)
\( x - \bar{x} = 210 - 181.5 = 28.5 \)
\( y - \bar{y} = 425 - 345.1667 = 79.8333 \)
\( (x - \bar{x})(y - \bar{y}) = (28.5)(79.8333) \approx 2275.25 \)
\( (x - \bar{x})^2 = (28.5)^2 = 812.25 \)
\( (y - \bar{y})^2 = (79.8333)^2 \approx 6373.0 \)

6. \( x = 243 \), \( y = 374 \)

Answer:

Step1: List the data points

We have the following data points for \( x \) (calories) and \( y \) (sales):
\( (121, 114) \), \( (145, 240) \), \( (181, 461) \), \( (189, 457) \), \( (210, 425) \), \( (243, 374) \)

Step2: Calculate necessary sums

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

• \( \sum x = 121 + 145 + 181 + 189 + 210 + 243 = 1089 \)
• \( \sum y = 114 + 240 + 461 + 457 + 425 + 374 = 2071 \)
• \( \sum xy = (121\times114)+(145\times240)+(181\times461)+(189\times457)+(210\times425)+(243\times374) \)

\( = 13794 + 34800 + 83441 + 86373 + 89250 + 90882 = 398540 \)

• \( \sum x^2 = 121^2 + 145^2 + 181^2 + 189^2 + 210^2 + 243^2 \)

\( = 14641 + 21025 + 32761 + 35721 + 44100 + 59049 = 207297 \)

• \( \sum y^2 = 114^2 + 240^2 + 461^2 + 457^2 + 425^2 + 374^2 \)

\( = 12996 + 57600 + 212521 + 208849 + 180625 + 140076 = 812667 \)

Step3: Use the correlation coefficient formula

The formula for the 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 \( n = 6 \) (number of data points)

First, calculate the numerator:
\( n\sum xy - \sum x \sum y = 6\times398540 - 1089\times2071 \)
\( = 2391240 - 2255319 = 135921 \)

Then, calculate the denominator:

• \( n\sum x^2 - (\sum x)^2 = 6\times207297 - 1089^2 \)

\( = 1243782 - 1185921 = 57861 \)

• \( n\sum y^2 - (\sum y)^2 = 6\times812667 - 2071^2 \)

\( = 4876002 - 4289041 = 586961 \)

• Now, the denominator is \( \sqrt{57861\times586961} \)

\( \sqrt{57861\times586961} \approx \sqrt{34060000000} \approx 184553.51 \) (approximate value, more accurately calculated as follows)
First, \( 57861\times586961 = 57861\times586961 \approx 34060000000 \) (but we calculate it precisely)
\( 57861\times586961 = 57861\times(500000 + 80000 + 6000 + 900 + 60 + 1) \)
\( = 57861\times500000 + 57861\times80000 + 57861\times6000 + 57861\times900 + 57861\times60 + 57861\times1 \)
\( = 28930500000 + 4628880000 + 347166000 + 52074900 + 3471660 + 57861 \)
\( = 28930500000 + 4628880000 = 33559380000 \)
\( 33559380000 + 347166000 = 33906546000 \)
\( 33906546000 + 52074900 = 33958620900 \)
\( 33958620900 + 3471660 = 33962092560 \)
\( 33962092560 + 57861 = 33962150421 \)
Then, \( \sqrt{33962150421} \approx 184288.23 \) (more accurate)

Now, \( r = \frac{135921}{184288.23} \approx 0.737 \)

Wait, let's recalculate the denominator more accurately:

First, \( n\sum x^2 - (\sum x)^2 = 6*207297 - 1089^2 = 1243782 - 1185921 = 57861 \)

\( n\sum y^2 - (\sum y)^2 = 6*812667 - 2071^2 = 4876002 - 4289041 = 586961 \)

Now, \( 57861*586961 = 57861*586961 \)

Let's compute \( 57861*586961 \):

First, note that \( 57861 = 3*19287 = 3*3*6429 = 3*3*3*2143 \)

\( 586961 \) is a prime? Maybe not, but we can use a calculator approach.

Alternatively, use the formula step by step with more precision.

Let's recalculate the numerator and denominator with more precision.

\( n = 6 \)

\( \sum x = 1089 \), \( \sum y = 2071 \)

\( \sum xy = 398540 \)

\( \sum x^2 = 207297 \)

\( \sum y^2 = 812667 \)

Numerator: \( 6398540 - 10892071 = 2391240 - (1089*2071) \)

Calculate \( 1089*2071 \):

\( 1000*2071 = 2071000 \)

\( 80*2071 = 165680 \)

\( 9*2071 = 18639 \)

Sum: \( 2071000 + 165680 = 2236680 + 18639 = 2255319 \)

So numerator: \( 2391240 - 2255319 = 135921 \)

Denominator part 1: \( n\sum x^2 - (\sum x)^2 = 6*207297 - 1089^2 = 1243782 - 1185921 = 57861 \)

Denominator part 2: \( n\sum y^2 - (\sum y)^2 = 6*812667 - 2071^2 = 4876002 - 4289041 = 586961 \)

Now, the denominator is \( \sqrt{57861 * 586961} \)

Calculate \( 57861 * 586961 \):

\( 57861 586961 = 57861 (586000 + 961) = 57861586000 + 57861961 \)

\( 57861*586000 = 57861*586*1000 \)

\( 57861586 = 57861(500 + 80 + 6) = 57861500 + 5786180 + 57861*6 \)

\( = 28930500 + 4628880 + 347166 = 28930500 + 4628880 = 33559380 + 347166 = 33906546 \)

So \( 57861*586000 = 33906546*1000 = 33906546000 \)

\( 57861961 = 57861(900 + 60 + 1) = 57861900 + 5786160 + 57861*1 \)

\( = 52074900 + 3471660 + 57861 = 52074900 + 3471660 = 55546560 + 57861 = 55604421 \)

So total \( 57861*586961 = 33906546000 + 55604421 = 33962150421 \)

Now, \( \sqrt{33962150421} \approx 184288.23 \) (using a calculator, \( \sqrt{33962150421} \approx 184288.23 \))

Now, \( r = \frac{135921}{184288.23} \approx 0.737 \) (rounded to the nearest thousandth)

Wait, but let's check with a calculator for more accuracy.

Alternatively, use the formula in a calculator-like approach:

The correlation coefficient formula can also be calculated using the following steps:

First, calculate the mean of \( x \), \( \bar{x} = \frac{\sum x}{n} = \frac{1089}{6} = 181.5 \)

Mean of \( y \), \( \bar{y} = \frac{\sum y}{n} = \frac{2071}{6} \approx 345.1667 \)

Then, calculate the numerator as \( \sum (x - \bar{x})(y - \bar{y}) \)

And the denominator as \( \sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2} \)

Let's calculate \( (x - \bar{x}) \) and \( (y - \bar{y}) \) for each data point:

1. \( x = 121 \), \( y = 114 \)

\( x - \bar{x} = 121 - 181.5 = -60.5 \)
\( y - \bar{y} = 114 - 345.1667 = -231.1667 \)
\( (x - \bar{x})(y - \bar{y}) = (-60.5)(-231.1667) \approx 13985.595 \)
\( (x - \bar{x})^2 = (-60.5)^2 = 3660.25 \)
\( (y - \bar{y})^2 = (-231.1667)^2 \approx 53438.0 \)

1. \( x = 145 \), \( y = 240 \)

\( x - \bar{x} = 145 - 181.5 = -36.5 \)
\( y - \bar{y} = 240 - 345.1667 = -105.1667 \)
\( (x - \bar{x})(y - \bar{y}) = (-36.5)(-105.1667) \approx 3848.585 \)
\( (x - \bar{x})^2 = (-36.5)^2 = 1332.25 \)
\( (y - \bar{y})^2 = (-105.1667)^2 \approx 11059.0 \)

1. \( x = 181 \), \( y = 461 \)

\( x - \bar{x} = 181 - 181.5 = -0.5 \)
\( y - \bar{y} = 461 - 345.1667 = 115.8333 \)
\( (x - \bar{x})(y - \bar{y}) = (-0.5)(115.8333) \approx -57.9167 \)
\( (x - \bar{x})^2 = (-0.5)^2 = 0.25 \)
\( (y - \bar{y})^2 = (115.8333)^2 \approx 13417.0 \)

1. \( x = 189 \), \( y = 457 \)

\( x - \bar{x} = 189 - 181.5 = 7.5 \)
\( y - \bar{y} = 457 - 345.1667 = 111.8333 \)
\( (x - \bar{x})(y - \bar{y}) = (7.5)(111.8333) \approx 838.75 \)
\( (x - \bar{x})^2 = (7.5)^2 = 56.25 \)
\( (y - \bar{y})^2 = (111.8333)^2 \approx 12507.0 \)

1. \( x = 210 \), \( y = 425 \)

\( x - \bar{x} = 210 - 181.5 = 28.5 \)
\( y - \bar{y} = 425 - 345.1667 = 79.8333 \)
\( (x - \bar{x})(y - \bar{y}) = (28.5)(79.8333) \approx 2275.25 \)
\( (x - \bar{x})^2 = (28.5)^2 = 812.25 \)
\( (y - \bar{y})^2 = (79.8333)^2 \approx 6373.0 \)

1. \( x = 243 \), \( y = 374 \)