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The table shows the battery lives, in hours, of ten Brand [tex]$A$[/tex] batteries and ten Brand [tex]$B$[/tex] batteries.

Battery Life (hours)
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Brand A & 22.5 & 17.0 & 21.0 & 23.0 & 22.0 & 18.5 & 22.5 & 20.0 & 19.0 & 23.0 \\
\hline
Brand B & 20.0 & 19.5 & 20.5 & 16.5 & 14.0 & 17.0 & 11.0 & 19.5 & 21.0 & 12.0 \\
\hline
\end{tabular}

Which would be the best measure of variability to use to compare the data?

A. Only Brand A data is symmetric, so standard deviation is the best measure to compare variability.
B. Only Brand B data is symmetric, so the median is the best measure to compare variability.
C. Both distributions are symmetric, so the mean is the best measure to compare variability.
D. Both distributions are skewed left, so the interquartile range is the best measure to compare variability.



Answer :

GinnyAnswer
To determine the best measure of variability to use to compare the data for the battery lives of Brand A and Brand B, let's follow these steps:

1. Calculating the mean:
- For Brand A: The mean battery life is [tex]\( \overline{X}_A = 20.85 \)[/tex] hours.
- For Brand B: The mean battery life is [tex]\( \overline{X}_B = 17.1 \)[/tex] hours.

2. Calculating the median:
- For Brand A: The median battery life is [tex]\( \text{Median}_A = 21.5 \)[/tex] hours.
- For Brand B: The median battery life is [tex]\( \text{Median}_B = 18.25 \)[/tex] hours.

3. Determining skewness:
- For Brand A: The skewness can be approximated by [tex]\( \text{Skewness}_A = | \overline{X}_A - \text{Median}_A | = | 20.85 - 21.5 | = 0.65 \)[/tex].
- For Brand B: The skewness can be approximated by [tex]\( \text{Skewness}_B = | \overline{X}_B - \text{Median}_B | = | 17.1 - 18.25 | = 1.15 \)[/tex].

4. Evaluating the symmetry:
- Since both skewness values for Brand A (0.65) and Brand B (1.15) are not close to 0, we can infer that both distributions are skewed.

5. Considering the best measure of variability:
- When distributions are symmetric, the standard deviation is often used to represent the variability.
- When distributions are skewed, the interquartile range (IQR) is a better measure of variability because it is not affected by extreme values or skewness.

Given that:
- The skewness for Brand A is 0.65.
- The skewness for Brand B is 1.15.
- These values indicate that both distributions are skewed.

Thus, the best measure of variability to compare the data for these two brands is the interquartile range (IQR).

Conclusion:
"Both distributions are skewed left, so the interquartile range is the best measure to compare variability."