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Gabe is the human resources manager for the Advanced Scientific Research Lab. He has to record the heights (in centimeters) and weights (in pounds) for each of the scientists in the lab.

[tex]\[
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline Height (cm) & 178 & 163 & 174 & 186 & 154 & 167 & 167 & 181 & 159 & 165 & 177 & 191 & 158 \\
\hline Weight (lbs) & 157 & 163 & 190 & 187 & 183 & 173 & 184 & 189 & 193 & 192 & 177 & 173 & 168 \\
\hline
\end{tabular}
\][/tex]

What is the shape of the height and weight distribution?

A. The height and weight distribution exhibit a negative and a positive skew, respectively.
B. Both the height and weight distribution exhibit a positive skew.
C. Both the height and weight distribution exhibit a negative skew.
D. Both the height and weight distribution are symmetric about the mean.
E. The height and weight distribution exhibit a positive and a negative skew, respectively.



Answer :

GinnyAnswer
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]

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