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Sagot :
To determine the shape of the height and weight distributions, we need to calculate the mean and median for both distributions and compare them.
### Heights:
- Data: 178, 163, 174, 186, 154, 167, 167, 181, 159, 165, 177, 191, 158
- Mean (Average): The mean of the height distribution is calculated by adding all the heights together and dividing by the number of data points.
[tex]\[ \text{Mean of heights} = \frac{178 + 163 + 174 + 186 + 154 + 167 + 167 + 181 + 159 + 165 + 177 + 191 + 158}{13} \approx 170.77 \][/tex]
- Median: The median is the middle value when the data is arranged in ascending order. For the height distribution:
Arranged data: 154, 158, 159, 163, 165, 167, 167, 174, 177, 178, 181, 186, 191
[tex]\[ \text{Median of heights} = 167.0 \][/tex]
#### Comparing Mean and Median for Heights:
- If the mean is greater than the median, the distribution is positively skewed.
- If the mean is less than the median, the distribution is negatively skewed.
- If the mean is equal to the median, the distribution is symmetric.
Since [tex]\( 170.77 > 167.0 \)[/tex], the height distribution is positively skewed.
### Weights:
- Data: 157, 163, 190, 187, 183, 173, 184, 189, 193, 192, 177, 173, 168
- Mean (Average): The mean of the weight distribution is calculated similarly by adding all the weights and dividing by the number of data points.
[tex]\[ \text{Mean of weights} = \frac{157 + 163 + 190 + 187 + 183 + 173 + 184 + 189 + 193 + 192 + 177 + 173 + 168}{13} \approx 179.15 \][/tex]
- Median: The median is the middle value when the data is arranged in ascending order. For the weight distribution:
Arranged data: 157, 163, 168, 173, 173, 177, 183, 184, 187, 189, 190, 192, 193
[tex]\[ \text{Median of weights} = 183.0 \][/tex]
#### Comparing Mean and Median for Weights:
- If the mean is greater than the median, the distribution is positively skewed.
- If the mean is less than the median, the distribution is negatively skewed.
- If the mean is equal to the median, the distribution is symmetric.
Since [tex]\( 179.15 < 183.0 \)[/tex], the weight distribution is negatively skewed.
### Conclusion:
- The height distribution is positively skewed.
- The weight distribution is negatively skewed.
Therefore, the correct answer is:
[tex]\[ \boxed{E. \text{The height and weight distribution exhibit a positive and a negative skew, respectively.}} \][/tex]
### Heights:
- Data: 178, 163, 174, 186, 154, 167, 167, 181, 159, 165, 177, 191, 158
- Mean (Average): The mean of the height distribution is calculated by adding all the heights together and dividing by the number of data points.
[tex]\[ \text{Mean of heights} = \frac{178 + 163 + 174 + 186 + 154 + 167 + 167 + 181 + 159 + 165 + 177 + 191 + 158}{13} \approx 170.77 \][/tex]
- Median: The median is the middle value when the data is arranged in ascending order. For the height distribution:
Arranged data: 154, 158, 159, 163, 165, 167, 167, 174, 177, 178, 181, 186, 191
[tex]\[ \text{Median of heights} = 167.0 \][/tex]
#### Comparing Mean and Median for Heights:
- If the mean is greater than the median, the distribution is positively skewed.
- If the mean is less than the median, the distribution is negatively skewed.
- If the mean is equal to the median, the distribution is symmetric.
Since [tex]\( 170.77 > 167.0 \)[/tex], the height distribution is positively skewed.
### Weights:
- Data: 157, 163, 190, 187, 183, 173, 184, 189, 193, 192, 177, 173, 168
- Mean (Average): The mean of the weight distribution is calculated similarly by adding all the weights and dividing by the number of data points.
[tex]\[ \text{Mean of weights} = \frac{157 + 163 + 190 + 187 + 183 + 173 + 184 + 189 + 193 + 192 + 177 + 173 + 168}{13} \approx 179.15 \][/tex]
- Median: The median is the middle value when the data is arranged in ascending order. For the weight distribution:
Arranged data: 157, 163, 168, 173, 173, 177, 183, 184, 187, 189, 190, 192, 193
[tex]\[ \text{Median of weights} = 183.0 \][/tex]
#### Comparing Mean and Median for Weights:
- If the mean is greater than the median, the distribution is positively skewed.
- If the mean is less than the median, the distribution is negatively skewed.
- If the mean is equal to the median, the distribution is symmetric.
Since [tex]\( 179.15 < 183.0 \)[/tex], the weight distribution is negatively skewed.
### Conclusion:
- The height distribution is positively skewed.
- The weight distribution is negatively skewed.
Therefore, the correct answer is:
[tex]\[ \boxed{E. \text{The height and weight distribution exhibit a positive and a negative skew, respectively.}} \][/tex]
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