On her daily homework assignments, Qinna has earned the maximum score of $10$ on $15$ out of $40$ days. The mode of her $40$ scores is $7$ and her median score is $9$. What is the least that her arithmetic mean could be? Express your answer as a decimal to the nearest tenth.

Leafie Dec 28, 2020

#1**+1 **

Not sure about this... But here's my rational

We know that we have \(15\) scores of \(10\).

Since the mode is \(7\) we need to have \(16\) scores of \(7\) (and no more)

The median is \(9\), so the \(20\) and \(21st\) scores must be \(9...\) to keep the average as small as possible.

And the other \(3\) scores need to be as small as possible so let them \(=1\)

We have

\(1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7 ,7, 7, 9 ,9, 9, 9, 9, 9,10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10\)

\(\text{Median} = 9\)

\(\text{Mode} = 7\)

\(\text{Mean} = 7.975 = 8\)

Credit to CPhill; https://web2.0calc.com/questions/basic-stats_2#r1

P.S. I changed up CPhill's answer a little, so please give me a little credit also!

cryptoaops Dec 29, 2020