To determine the shapes of the height and weight distributions, we need to evaluate their skewness. Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.
Skewness can be interpreted as follows:
- If skewness is less than 0, the distribution is negatively skewed.
- If skewness is approximately 0, the distribution is symmetric.
- If skewness is greater than 0, the distribution is positively skewed.
From our calculations, we have obtained the following results:
- The skewness for the height distribution is approximately 0.256.
- The skewness for the weight distribution is approximately -0.510.
Analyzing these values:
1. For the height distribution:
- The skewness is approximately 0.256, which is greater than 0. This indicates that the height distribution is positively skewed.
2. For the weight distribution:
- The skewness is approximately -0.510, which is less than 0. This indicates that the weight distribution is negatively skewed.
Hence, the shapes of the distributions are as follows:
- The height distribution is positively skewed.
- The weight distribution is negatively skewed.
Based on this analysis, the correct answer is:
E. The height and weight distribution exhibit a positive and a negative skew, respectively.
