Question

An online store receives customer satisfaction ratings between $0$ and $100$, inclusive. In the first $10$ ratings the store received, the average (arithmetic mean) of the ratings was $75$. What is the least value the store can receive for the ${11}^{th}$ rating and still be able to have an average of at least $85$ for the first $20$ ratings?

• A. $40$
• B. $50$
• C. $30$
• D. $60$

Answer 1

Answer: (B)

$50$

The mean of a data set is the sum of the values in the data set divided by the number of values in the data set. The mean of $75$ was obtained by finding the sum of the first $10$ ratings and dividing by $10$.

Thus, the sum of the first $10$ ratings was $750$. In order for the mean of the first $20$ ratings to be at least $85$, the sum of the first $20$ ratings must be at least $85\left(20\right)=1700$.

Therefore, the sum of the next $10$ ratings must be at least $1700-750=950$. The maximum rating is $100$, so the maximum possible value of the sum of the ${12}^{th}$ through ${20}^{th}$ ratings is $9\left(100\right)=900$.

Therefore, for the store to be able to have an average of at least $85$ for the first $20$ ratings, the least possible value for the ${11}^{th}$ rating is $950-900=50$.