Question
Dozen Eggs:
1.94
1.88
1.83
1.77
1.65
2.01
1.80
1.96
Gallon of Milk
3.47
3.52
3.58
3.54
3.47
3.52
3.47
3.58
Compute the correlation co-efficient. Round to 3 decimal places
1.94
1.88
1.83
1.77
1.65
2.01
1.80
1.96
Gallon of Milk
3.47
3.52
3.58
3.54
3.47
3.52
3.47
3.58
Compute the correlation co-efficient. Round to 3 decimal places
Answers
GPT-4o mini
To compute the correlation coefficient (Pearson's r) for the given data sets (dozen eggs and gallon of milk), we will follow these steps:
1. **Calculate the means of each data set.**
2. **Calculate the deviations from the mean for each data point.**
3. **Compute the products of these deviations.**
4. **Calculate the sum of the products, the sum of the squares of the deviations for each data set.**
5. **Use the formula for Pearson's correlation coefficient.**
Let's denote:
- \(X\) as the prices of a dozen eggs,
- \(Y\) as the prices of a gallon of milk.
### Given data:
- **Dozen Eggs (X):** 1.94, 1.88, 1.83, 1.77, 1.65, 2.01, 1.80, 1.96
- **Gallon of Milk (Y):** 3.47, 3.52, 3.58, 3.54, 3.47, 3.52, 3.47, 3.58
### Step 1: Calculate the means
\[
\text{Mean of } X = \frac{1.94 + 1.88 + 1.83 + 1.77 + 1.65 + 2.01 + 1.80 + 1.96}{8} = \frac{13.84}{8} = 1.73
\]
\[
\text{Mean of } Y = \frac{3.47 + 3.52 + 3.58 + 3.54 + 3.47 + 3.52 + 3.47 + 3.58}{8} = \frac{28.13}{8} = 3.51625
\]
### Step 2: Calculate deviations and products
Now we need to calculate the deviations from the mean for each value and then multiply them.
\[
\text{Deviations for } X:
\begin{align*}
1.94 - 1.73 &= 0.21 \\
1.88 - 1.73 &= 0.15 \\
1.83 - 1.73 &= 0.10 \\
1.77 - 1.73 &= 0.04 \\
1.65 - 1.73 &= -0.08 \\
2.01 - 1.73 &= 0.28 \\
1.80 - 1.73 &= 0.07 \\
1.96 - 1.73 &= 0.23 \\
\end{align*}
\]
\[
\text{Deviations for } Y:
\begin{align*}
3.47 - 3.51625 &= -0.04625 \\
3.52 - 3.51625 &= 0.00375 \\
3.58 - 3.51625 &= 0.06375 \\
3.54 - 3.51625 &= 0.02375 \\
3.47 - 3.51625 &= -0.04625 \\
3.52 - 3.51625 &= 0.00375 \\
3.47 - 3.51625 &= -0.04625 \\
3.58 - 3.51625 &= 0.06375 \\
\end{align*}
\]
Now calculate \( (X_i - \bar{X})(Y_i - \bar{Y}) \):
- For \(X = 1.94\) and \(Y = 3.47:\) \(0.21 \cdot -0.04625 = -0.00973375\)
- For \(X = 1.88\) and \(Y = 3.52:\) \(0.15 \cdot 0.00375 = 0.0005625\)
- For \(X = 1.83\) and \(Y = 3.58:\) \(0.10 \cdot 0.06375 = 0.006375\)
- For \(X = 1.77\) and \(Y = 3.54:\) \(0.04 \cdot 0.02375 = 0.00095\)
- For \(X = 1.65\) and \(Y = 3.47:\) \(-0.08 \cdot -0.04625 = 0.00371\)
- For \(X = 2.01\) and \(Y = 3.52:\) \(0.28 \cdot 0.00375 = 0.00105\)
- For \(X = 1.80\) and \(Y = 3.47:\) \(0.07 \cdot -0.04625 = -0.0032375\)
- For \(X = 1.96\) and \(Y = 3.58:\) \(0.23 \cdot 0.06375 = 0.0146625\)
### Step 3: Sum these values
Now summing these products:
\[
S_{XY} = -0.00973375 + 0.0005625 + 0.006375 + 0.00095 + 0.00371 + 0.00105 - 0.0032375 + 0.0146625 \approx 0.01364375
\]
### Step 4: Calculate the sum of squares
Next, we will compute \( \sum (X_i - \bar{X})^2 \) and \( \sum (Y_i - \bar{Y})^2 \):
For X:
\[
\sum (X_i - \bar{X})^2 = 0.21^2 + 0.15^2 + 0.10^2 + 0.04^2 + (-0.08)^2 + 0.28^2 + 0.07^2 + 0.23^2 \approx 0.0441 + 0.0225 + 0.01 + 0.0016 + 0.0064 + 0.0784 + 0.0049 + 0.0529 \approx 0.2209
\]
For Y:
\[
\sum (Y_i - \bar{Y})^2 = (-0.04625)^2 + (0.00375)^2 + (0.06375)^2 + (0.02375)^2 + (-0.04625)^2 + (0.00375)^2 + (-0.04625)^2 + (0.06375)^2 \approx 0.0021380625 + 0.0000140625 + 0.00405625 + 0.0005640625 + 0.0021380625 + 0.0000140625 + 0.0021380625 + 0.00405625 \approx 0.0040205
\]
### Step 5: Compute the correlation coefficient
Now we can calculate the correlation coefficient using the formula:
\[
r = \frac{S_{XY}}{\sqrt{(\sum (X_i - \bar{X})^2)(\sum (Y_i - \bar{Y})^2)}}
\]
Substituting the values we calculated:
\[
r = \frac{0.01364375}{\sqrt{0.2209 \cdot 0.0003940625}} \approx \frac{0.01364375}{\sqrt{0.000087074}}
\]
Calculating the denominator:
\[
\sqrt{0.087074} \approx 0.295064
\]
Thus,
\[
r \approx \frac{0.01364375}{0.295064} \approx 0.0462
\]
After rounding to 3 decimal places, the correlation coefficient is approximately:
\[
\boxed{0.046}
\]
1. **Calculate the means of each data set.**
2. **Calculate the deviations from the mean for each data point.**
3. **Compute the products of these deviations.**
4. **Calculate the sum of the products, the sum of the squares of the deviations for each data set.**
5. **Use the formula for Pearson's correlation coefficient.**
Let's denote:
- \(X\) as the prices of a dozen eggs,
- \(Y\) as the prices of a gallon of milk.
### Given data:
- **Dozen Eggs (X):** 1.94, 1.88, 1.83, 1.77, 1.65, 2.01, 1.80, 1.96
- **Gallon of Milk (Y):** 3.47, 3.52, 3.58, 3.54, 3.47, 3.52, 3.47, 3.58
### Step 1: Calculate the means
\[
\text{Mean of } X = \frac{1.94 + 1.88 + 1.83 + 1.77 + 1.65 + 2.01 + 1.80 + 1.96}{8} = \frac{13.84}{8} = 1.73
\]
\[
\text{Mean of } Y = \frac{3.47 + 3.52 + 3.58 + 3.54 + 3.47 + 3.52 + 3.47 + 3.58}{8} = \frac{28.13}{8} = 3.51625
\]
### Step 2: Calculate deviations and products
Now we need to calculate the deviations from the mean for each value and then multiply them.
\[
\text{Deviations for } X:
\begin{align*}
1.94 - 1.73 &= 0.21 \\
1.88 - 1.73 &= 0.15 \\
1.83 - 1.73 &= 0.10 \\
1.77 - 1.73 &= 0.04 \\
1.65 - 1.73 &= -0.08 \\
2.01 - 1.73 &= 0.28 \\
1.80 - 1.73 &= 0.07 \\
1.96 - 1.73 &= 0.23 \\
\end{align*}
\]
\[
\text{Deviations for } Y:
\begin{align*}
3.47 - 3.51625 &= -0.04625 \\
3.52 - 3.51625 &= 0.00375 \\
3.58 - 3.51625 &= 0.06375 \\
3.54 - 3.51625 &= 0.02375 \\
3.47 - 3.51625 &= -0.04625 \\
3.52 - 3.51625 &= 0.00375 \\
3.47 - 3.51625 &= -0.04625 \\
3.58 - 3.51625 &= 0.06375 \\
\end{align*}
\]
Now calculate \( (X_i - \bar{X})(Y_i - \bar{Y}) \):
- For \(X = 1.94\) and \(Y = 3.47:\) \(0.21 \cdot -0.04625 = -0.00973375\)
- For \(X = 1.88\) and \(Y = 3.52:\) \(0.15 \cdot 0.00375 = 0.0005625\)
- For \(X = 1.83\) and \(Y = 3.58:\) \(0.10 \cdot 0.06375 = 0.006375\)
- For \(X = 1.77\) and \(Y = 3.54:\) \(0.04 \cdot 0.02375 = 0.00095\)
- For \(X = 1.65\) and \(Y = 3.47:\) \(-0.08 \cdot -0.04625 = 0.00371\)
- For \(X = 2.01\) and \(Y = 3.52:\) \(0.28 \cdot 0.00375 = 0.00105\)
- For \(X = 1.80\) and \(Y = 3.47:\) \(0.07 \cdot -0.04625 = -0.0032375\)
- For \(X = 1.96\) and \(Y = 3.58:\) \(0.23 \cdot 0.06375 = 0.0146625\)
### Step 3: Sum these values
Now summing these products:
\[
S_{XY} = -0.00973375 + 0.0005625 + 0.006375 + 0.00095 + 0.00371 + 0.00105 - 0.0032375 + 0.0146625 \approx 0.01364375
\]
### Step 4: Calculate the sum of squares
Next, we will compute \( \sum (X_i - \bar{X})^2 \) and \( \sum (Y_i - \bar{Y})^2 \):
For X:
\[
\sum (X_i - \bar{X})^2 = 0.21^2 + 0.15^2 + 0.10^2 + 0.04^2 + (-0.08)^2 + 0.28^2 + 0.07^2 + 0.23^2 \approx 0.0441 + 0.0225 + 0.01 + 0.0016 + 0.0064 + 0.0784 + 0.0049 + 0.0529 \approx 0.2209
\]
For Y:
\[
\sum (Y_i - \bar{Y})^2 = (-0.04625)^2 + (0.00375)^2 + (0.06375)^2 + (0.02375)^2 + (-0.04625)^2 + (0.00375)^2 + (-0.04625)^2 + (0.06375)^2 \approx 0.0021380625 + 0.0000140625 + 0.00405625 + 0.0005640625 + 0.0021380625 + 0.0000140625 + 0.0021380625 + 0.00405625 \approx 0.0040205
\]
### Step 5: Compute the correlation coefficient
Now we can calculate the correlation coefficient using the formula:
\[
r = \frac{S_{XY}}{\sqrt{(\sum (X_i - \bar{X})^2)(\sum (Y_i - \bar{Y})^2)}}
\]
Substituting the values we calculated:
\[
r = \frac{0.01364375}{\sqrt{0.2209 \cdot 0.0003940625}} \approx \frac{0.01364375}{\sqrt{0.000087074}}
\]
Calculating the denominator:
\[
\sqrt{0.087074} \approx 0.295064
\]
Thus,
\[
r \approx \frac{0.01364375}{0.295064} \approx 0.0462
\]
After rounding to 3 decimal places, the correlation coefficient is approximately:
\[
\boxed{0.046}
\]