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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 8 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.

Jul 2, 2022

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I am assuming the score goes from 0 - 10.

Because 8 is the mode, it must occur the most often. We also know that there are 15 10's in the data set, there must be at least 16 8's in the data set.

To minimize the mean, we need to include as many 0's as possible and limit the amount of "high" numbers. This means there will be exactly 16 8's

Also note that because we know the median is 9, the 20th and the 21st numbers must both be 9.

If we order them in a list (lowest to highest), we have 15 perfect scores (26th to the 40th). We will also have exactly 7 9's (20th to the 25th), so the median will be 9. We also will have 16 8's (4th to 19th), to make the mode 8. This means that the remaining 3 spots will be 0's (1st to 3rd), representing the days Qinna was too lazy to do her homework.

So, the mean is $$((10 \times 15) + (7 \times 9) + (16 \times 8)) \div 40 \approx \color{brown}\boxed{8.5}$$

Jul 2, 2022