To determine the correlation between age and weight based on the given data, we need to look at how the variables are related to each other.
Given data:
- Age in weeks: 1, 2, 3, 3, 4, 4, 6, 8, 9, 9
- Weight in lbs: 7.5, 7.25, 8.2, 7.95, 8.0, 9.75, 9.25, 8.9, 9.85, 10.0
Here’s the step-by-step solution:
1. Arrange the Data: List the ages and corresponding weights clearly.
- Age: [1, 2, 3, 3, 4, 4, 6, 8, 9, 9]
- Weight: [7.5, 7.25, 8.2, 7.95, 8.0, 9.75, 9.25, 8.9, 9.85, 10.0]
2. Compute the Mean: Find the mean (average) of the age and weight values.
- Mean of Age = [tex]\((1 + 2 + 3 + 3 + 4 + 4 + 6 + 8 + 9 + 9) / 10 = 4.9\)[/tex]
- Mean of Weight = [tex]\((7.5 + 7.25 + 8.2 + 7.95 + 8.0 + 9.75 + 9.25 + 8.9 + 9.85 + 10.0) / 10 = 8.665\)[/tex]
3. Calculate the Covariance and Standard Deviations:
- We use covariance to understand how two variables vary together.
- Standard deviation measures the amount of variation or dispersion of a set of values.
Without going into the complexities of the formula, we make use of the correlation coefficient formula:
[tex]\[ r = \frac{{\text{cov}(X, Y)}}{{\sigma_X \sigma_Y}} \][/tex]
where [tex]\( r \)[/tex] is the correlation coefficient, cov(X, Y) is the covariance of age and weight, and [tex]\(\sigma_X\)[/tex] and [tex]\(\sigma_Y\)[/tex] are the standard deviations of age and weight respectively.
4. Compute the Correlation Coefficient:
- After computing the covariance and standard deviations, plug in the values to get the correlation coefficient.
- For our data set, we ultimately find:
[tex]\[ r = 0.8311 \][/tex]
5. Interpret the Correlation Coefficient:
- The correlation coefficient [tex]\( r = 0.8311 \)[/tex] indicates a strong positive correlation.
Based on the numerical result, we conclude:
- The correlation between age and weight in the given data set is positive.
Therefore, the correct answer is: positive.
This means that as the age of the babies increases, their weight tends to also increase.
