The table shows the fat content and calories for the burgers at a fast food chain.

Question

Fat (g) 25 44 63 32 37 20 11 52
Calories 590 830 1080 680 750 420 310 820
How strong is the correlation between fat content (g) and calories? State what you need to calculate to determine the correlation and interpret the results.

Answer 1

To determine the correlation between fat content (g) and calories for the burgers at the fast food chain, we will calculate the Pearson correlation coefficient (r). This coefficient measures the strength and direction of the linear relationship between two variables.

Steps to Calculate the Correlation Coefficient:

1. Collect Data:

   • Fat content (g): [25, 44, 63, 32, 37, 20, 11, 52]
   • Calories: [590, 830, 1080, 680, 750, 420, 310, 820]

2. Calculate Means:

   • Find the mean of the fat content (g).
   • Find the mean of calories.

3. Compute the Covariance:

   • Calculate the covariance between the two variables. The formula is: \[ \text{Cov}(X, Y) = \frac{\sum (x_i - \bar{X})(y_i - \bar{Y})}{n} \] Where \(X\) is fat content, \(Y\) is calories, \(x_i\) and \(y_i\) are the individual measurements, \(\bar{X}\) and \(\bar{Y}\) are the means, and \(n\) is the number of data points.

4. Calculate the Standard Deviations:

   • Find the standard deviation for fat content and for calories using the formulas: \[ s_X = \sqrt{\frac{\sum (x_i - \bar{X})^2}{n-1}} \quad \text{and} \quad s_Y = \sqrt{\frac{\sum (y_i - \bar{Y})^2}{n-1}} \]

5. Calculate the Correlation Coefficient:

   • Finally, use the correlation formula: \[ r = \frac{\text{Cov}(X, Y)}{s_X s_Y} \]

Interpretation of the Results:

• The value of \(r\) ranges from -1 to 1.

  • \(r = 1\) indicates a perfect positive correlation (as one variable increases, the other also increases).
  • \(r = -1\) indicates a perfect negative correlation (as one variable increases, the other decreases).
  • \(r = 0\) suggests no linear correlation between the variables.

• Generally, we interpret:

  • \(0.7 < |r| \leq 1\): Strong correlation
  • \(0.3 < |r| \leq 0.7\): Moderate correlation
  • \(0 < |r| \leq 0.3\): Weak correlation

Calculating Correlation Coefficient (Example):

Using statistical software or manual calculation, after performing the calculations, you would find the correlation coefficient \(r\).

Example Conclusion:

Once calculated, you can interpret the strength of the relationship. For example, if \(r = 0.85\), you would conclude that there is a strong positive correlation between fat content and calorie content in the burgers. Conversely, if \(r = -0.5\), it would suggest a moderate negative correlation.

If you'd like, I can help walk through the specific calculations or present a calculated value for \(r\).

Answer 2

Please walk through specific calculations.

Answer 3

To calculate the Pearson correlation coefficient \( r \) between fat content (g) and calories, let's go through the steps methodically:

Step 1: Collect Data

• Fat content (g): \( X = [25, 44, 63, 32, 37, 20, 11, 52] \)
• Calories: \( Y = [590, 830, 1080, 680, 750, 420, 310, 820] \)

Step 2: Calculate the Means

1. Mean of Fat Content (\( \bar{X} \)): \[ \bar{X} = \frac{\sum X}{n} = \frac{25 + 44 + 63 + 32 + 37 + 20 + 11 + 52}{8} = \frac{284}{8} = 35.5 \]

2. Mean of Calories (\( \bar{Y} \)): \[ \bar{Y} = \frac{\sum Y}{n} = \frac{590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820}{8} = \frac{4580}{8} = 572.5 \]

Step 3: Compute the Covariance

We will use the formula: \[ \text{Cov}(X, Y) = \frac{\sum (x_i - \bar{X})(y_i - \bar{Y})}{n} \]

We'll calculate \( (x_i - \bar{X})(y_i - \bar{Y}) \) for each data point:

| i | \( x_i \) | \( y_i \) | \( x_i - \bar{X} \) | \( y_i - \bar{Y} \) | Product | |---|-------|-------|--------------------|--------------------|---------| | 1 | 25 | 590 | \( 25 - 35.5 = -10.5 \) | \( 590 - 572.5 = 17.5 \) | \( -10.5 \times 17.5 = -183.75 \) | | 2 | 44 | 830 | \( 44 - 35.5 = 8.5 \) | \( 830 - 572.5 = 257.5 \) | \( 8.5 \times 257.5 = 2198.75 \) | | 3 | 63 | 1080 | \( 63 - 35.5 = 27.5 \) | \( 1080 - 572.5 = 507.5 \)| \( 27.5 \times 507.5 = 13956.25 \) | | 4 | 32 | 680 | \( 32 - 35.5 = -3.5 \) | \( 680 - 572.5 = 107.5 \) | \( -3.5 \times 107.5 = -376.25 \) | | 5 | 37 | 750 | \( 37 - 35.5 = 1.5 \) | \( 750 - 572.5 = 177.5 \) | \( 1.5 \times 177.5 = 266.25 \) | | 6 | 20 | 420 | \( 20 - 35.5 = -15.5 \) | \( 420 - 572.5 = -152.5 \)| \( -15.5 \times -152.5 = 2363.75 \) | | 7 | 11 | 310 | \( 11 - 35.5 = -24.5 \) | \( 310 - 572.5 = -262.5 \)| \( -24.5 \times -262.5 = 6431.25 \) | | 8 | 52 | 820 | \( 52 - 35.5 = 16.5 \) | \( 820 - 572.5 = 247.5 \) | \( 16.5 \times 247.5 = 4083.75 \) |

Now sum the products: \[ \sum (x_i - \bar{X})(y_i - \bar{Y}) = -183.75 + 2198.75 + 13956.25 - 376.25 + 266.25 + 2363.75 + 6431.25 + 4083.75 = 20500 \]

Now, we calculate the covariance: \[ \text{Cov}(X, Y) = \frac{20500}{8} = 2562.5 \]

Step 4: Calculate the Standard Deviations

1. Standard Deviation of \( X \): \[ s_X = \sqrt{\frac{\sum (x_i - \bar{X})^2}{n-1}} \] Calculating \( (x_i - \bar{X})^2 \) for each: \[ \sum (x_i - \bar{X})^2 = (-10.5)^2 + (8.5)^2 + (27.5)^2 + (-3.5)^2 + (1.5)^2 + (-15.5)^2 + (-24.5)^2 + (16.5)^2 \] \[ = 110.25 + 72.25 + 756.25 + 12.25 + 2.25 + 240.25 + 600.25 + 272.25 = 2072.00 \] Now calculating \( s_X \): \[ s_X = \sqrt{\frac{2072.00}{7}} \approx \sqrt{296.00} \approx 17.2 \]

2. Standard Deviation of \( Y \): \[ s_Y = \sqrt{\frac{\sum (y_i - \bar{Y})^2}{n-1}} \] Calculating \( (y_i - \bar{Y})^2 \) for each: \[ \sum (y_i - \bar{Y})^2 = (17.5)^2 + (257.5)^2 + (507.5)^2 + (107.5)^2 + (177.5)^2 + (-152.5)^2 + (-262.5)^2 + (247.5)^2 \] \[ = 306.25 + 66406.25 + 257525.25 + 11556.25 + 31506.25 + 23256.25 + 68906.25 + 61256.25 = 418834 \] Now calculating \( s_Y \): \[ s_Y = \sqrt{\frac{418834}{7}} \approx \sqrt{59833.43} \approx 245.7 \]

Step 5: Calculate the Correlation Coefficient

Using the formula: \[ r = \frac{\text{Cov}(X, Y)}{s_X s_Y} = \frac{2562.5}{17.2 \times 245.7} \] Calculating \( s_X s_Y \): \[ s_X s_Y \approx 17.2 \times 245.7 \approx 4224.84 \] Calculating \( r \): \[ r = \frac{2562.5}{4224.84} \approx 0.606 \]

Interpretation of Results

The calculated correlation coefficient \( r \approx 0.606 \) indicates a moderate positive correlation between fat content and calories for the burgers at this fast food chain. This means that, on average, as the fat content in the burgers increases, the calorie content also tends to increase. However, the correlation is not perfect, indicating that other factors may also influence calorie content.