Question

1. a. calculate joeys mean quiz grade. b. joey has an unexcused absence for the 14th quiz, and he receives a score of zero. recalculate the mean and median. explain why the mean and median are so different now.

Explanation:

Step1: Recall the mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $x_{i}$ are the data - points and $n$ is the number of data - points. But we are not given the individual scores for part (a). Let's assume the scores of the first 13 quizzes are $x_1,x_2,\cdots,x_{13}$ and the sum of these scores is $S_{13}=\sum_{i = 1}^{13}x_{i}$, and the number of quizzes $n = 13$. The mean of the first 13 quizzes is $\bar{x}_1=\frac{S_{13}}{13}$.

Step2: Calculate the new mean for part (b)

For part (b), after the 14th quiz with a score of 0, the new sum of scores is $S_{14}=S_{13}+0 = S_{13}$, and the new number of quizzes is $n = 14$. So the new mean $\bar{x}_2=\frac{S_{13}}{14}$.
To find the median:

1. First, for the 13 - data - point set (in part (a)), if we order the scores from smallest to largest, the median is the 7th value (since $n = 13$ and the median position is $\frac{n + 1}{2}=\frac{13+1}{2}=7$).
2. For the 14 - data - point set (in part (b)), if we order the scores from smallest to largest, the median is the average of the 7th and 8th values (since $n = 14$ and the median position is $\frac{n}{2}=7$ and $\frac{n}{2}+1 = 8$, and the median $M=\frac{x_{7}+x_{8}}{2}$, where $x_{i}$ are the ordered scores).

The mean and median are different because the mean is affected by extreme values (in this case, the score of 0 is an extreme value relative to the other possible non - zero scores). The median is only based on the position of the data in the ordered set and is not affected by the magnitude of extreme values.

Answer:

a. Without the individual scores of the 13 quizzes, we cannot calculate the exact mean. The formula is $\bar{x}=\frac{\sum_{i = 1}^{13}x_{i}}{13}$.
b. Mean: $\frac{S_{13}}{14}$ (where $S_{13}$ is the sum of the first 13 scores). Median: average of the 7th and 8th ordered scores. The mean is affected by the extreme value (0), while the median is based on position and is less affected by extreme values.