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To calculate the weighted mean, you have to multiply each score by the corresponding percentage (expressed as a decimal value) and then add the results.
For the final grade, you know that the grade in each quiz counts for 10% of the final grade. So you have to multiply the score of each quiz by 0.10.
The final exam counts for 70% of the final grade, which means that you have to multiply the grade by 0.7.
[tex]\begin{gathered} \bar{X}=0.1\cdot90+0.1\cdot68+0.1\cdot80+0.7\cdot71 \\ \bar{X}=9+6.8+8+49.7 \\ \bar{X}=73.8 \end{gathered}[/tex]The weighted mean is equal to 73.8.
The letter grade corresponds to C.