Answer :
Certainly! Let's go through the process of finding the Pearson's correlation coefficient for the given data step-by-step.
### Step 1: Gather the Data
We have two sets of data:
- [tex]\( X \)[/tex] values: 45, 18, 63, 105, 0, 0, 111, 29
- [tex]\( Y \)[/tex] values: 79, 113, 59, 39, 219, 224, 131, 109
### Step 2: Calculate the Means
Calculate the mean of both sets of values.
Mean of [tex]\( X \)[/tex]:
[tex]\[
\text{Mean}_X = \frac{45 + 18 + 63 + 105 + 0 + 0 + 111 + 29}{8} = 46.375
\][/tex]
Mean of [tex]\( Y \)[/tex]:
[tex]\[
\text{Mean}_Y = \frac{79 + 113 + 59 + 39 + 219 + 224 + 131 + 109}{8} = 121.625
\][/tex]
### Step 3: Calculate the Covariance
The covariance shows how much two random variables vary together. Use the formula:
[tex]\[
\text{Covariance} = \frac{\sum{(x_i - \text{Mean}_X)(y_i - \text{Mean}_Y)}}{n}
\][/tex]
For our values, the covariance is:
[tex]\[
-1752.484375
\][/tex]
### Step 4: Calculate the Variances
Variance measures how far a set of numbers are spread out from their average value.
Variance of [tex]\( X \)[/tex]:
[tex]\[
\text{Variance}_X = \frac{\sum{(x_i - \text{Mean}_X)^2}}{n} = 1662.484375
\][/tex]
Variance of [tex]\( Y \)[/tex]:
[tex]\[
\text{Variance}_Y = \frac{\sum{(y_i - \text{Mean}_Y)^2}}{n} = 4106.234375
\][/tex]
### Step 5: Calculate Pearson's Correlation Coefficient
Pearson's correlation coefficient [tex]\( r \)[/tex] is calculated as follows:
[tex]\[
r = \frac{\text{Covariance}}{\sqrt{\text{Variance}_X \times \text{Variance}_Y}}
\][/tex]
Substituting the known values:
[tex]\[
r = \frac{-1752.484375}{\sqrt{1662.484375 \times 4106.234375}} = -0.6707389107098695
\][/tex]
### Conclusion
The Pearson's correlation coefficient for the given data is approximately [tex]\(-0.67\)[/tex]. This indicates a moderate negative relationship between the [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] values in the dataset.
### Step 1: Gather the Data
We have two sets of data:
- [tex]\( X \)[/tex] values: 45, 18, 63, 105, 0, 0, 111, 29
- [tex]\( Y \)[/tex] values: 79, 113, 59, 39, 219, 224, 131, 109
### Step 2: Calculate the Means
Calculate the mean of both sets of values.
Mean of [tex]\( X \)[/tex]:
[tex]\[
\text{Mean}_X = \frac{45 + 18 + 63 + 105 + 0 + 0 + 111 + 29}{8} = 46.375
\][/tex]
Mean of [tex]\( Y \)[/tex]:
[tex]\[
\text{Mean}_Y = \frac{79 + 113 + 59 + 39 + 219 + 224 + 131 + 109}{8} = 121.625
\][/tex]
### Step 3: Calculate the Covariance
The covariance shows how much two random variables vary together. Use the formula:
[tex]\[
\text{Covariance} = \frac{\sum{(x_i - \text{Mean}_X)(y_i - \text{Mean}_Y)}}{n}
\][/tex]
For our values, the covariance is:
[tex]\[
-1752.484375
\][/tex]
### Step 4: Calculate the Variances
Variance measures how far a set of numbers are spread out from their average value.
Variance of [tex]\( X \)[/tex]:
[tex]\[
\text{Variance}_X = \frac{\sum{(x_i - \text{Mean}_X)^2}}{n} = 1662.484375
\][/tex]
Variance of [tex]\( Y \)[/tex]:
[tex]\[
\text{Variance}_Y = \frac{\sum{(y_i - \text{Mean}_Y)^2}}{n} = 4106.234375
\][/tex]
### Step 5: Calculate Pearson's Correlation Coefficient
Pearson's correlation coefficient [tex]\( r \)[/tex] is calculated as follows:
[tex]\[
r = \frac{\text{Covariance}}{\sqrt{\text{Variance}_X \times \text{Variance}_Y}}
\][/tex]
Substituting the known values:
[tex]\[
r = \frac{-1752.484375}{\sqrt{1662.484375 \times 4106.234375}} = -0.6707389107098695
\][/tex]
### Conclusion
The Pearson's correlation coefficient for the given data is approximately [tex]\(-0.67\)[/tex]. This indicates a moderate negative relationship between the [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] values in the dataset.
