Question

1. on each of her exams this semester, emily has scored between 60 and 70. if she scores a 90 on her next exam, which of the following must be true? a. her mean score will decrease. b. her mean score will stay the same. c. her mean score will increase. d. her mean score may increase or decrease depending on how many exams she has taken.

Explanation:

Step1: Recall Mean Formula

The mean (average) score is calculated as $\frac{\text{Sum of all scores}}{\text{Number of exams}}$. Let the number of exams before the next one be $n$, and the sum of her previous scores be $S$, where each score is between 60 and 70, so $60n \leq S \leq 70n$.

Step2: Calculate New Mean

After the next exam (score = 90), the new sum is $S + 90$, and the new number of exams is $n + 1$. The new mean is $\frac{S + 90}{n + 1}$. Let's compare this to the old mean $\frac{S}{n}$.

Subtract the old mean from the new mean: $\frac{S + 90}{n + 1}-\frac{S}{n}=\frac{n(S + 90)-S(n + 1)}{n(n + 1)}=\frac{nS + 90n - Sn - S}{n(n + 1)}=\frac{90n - S}{n(n + 1)}$.

Since $S \leq 70n$, then $90n - S \geq 90n - 70n = 20n>0$ (for $n>0$). So the new mean minus the old mean is positive, meaning the new mean is greater than the old mean.

Answer:

C. Her mean score will increase.