To find the exact expression that represents the ratio of the difference of the two means to Sidney's mean absolute deviation, let's go through the steps:
1. Identify the means of Sidney and Phil:
- Mean of Sidney's grades ([tex]\( \text{mean}_{Sidney} \)[/tex]) = 82
- Mean of Phil's grades ([tex]\( \text{mean}_{Phil} \)[/tex]) = 78
2. Mean Absolute Deviations (MAD):
- MAD of Sidney's grades ([tex]\( \text{mad}_{Sidney} \)[/tex]) = 3.28
- MAD of Phil's grades ([tex]\( \text{mad}_{Phil} \)[/tex]) = 3.96
3. Calculate the difference between the means of Sidney's and Phil's grades:
[tex]\[
\text{Difference of means} = \text{mean}_{Sidney} - \text{mean}_{Phil}
\][/tex]
Substituting the values:
[tex]\[
\text{Difference of means} = 82 - 78 = 4
\][/tex]
4. Calculate the ratio of the difference of means to Sidney's mean absolute deviation:
[tex]\[
\text{Ratio} = \frac{\text{Difference of means}}{\text{mad}_{Sidney}}
\][/tex]
Substituting the values:
[tex]\[
\text{Ratio} = \frac{4}{3.28}
\][/tex]
Among the given expressions:
- [tex]\(\frac{82}{3.28}\)[/tex] does not represent our context as it directly divides Sidney's mean by his MAD.
- [tex]\(\frac{4}{0.68}\)[/tex] is not correct because it improperly uses the difference between the mean absolute deviations instead of the means.
- [tex]\(\frac{4}{3.28}\)[/tex] matches our calculation.
Therefore, the correct expression representing the ratio of the difference of the two means to Sidney's mean absolute deviation is:
[tex]\[
\boxed{\frac{4}{3.28}}
\][/tex]
