12 multiple-choice questions, progressively harder.
A student has scores of 787878, 858585, and 909090 on three tests. What must she score on a fourth test to have a mean of 858585?
Solution
Correct answer: B
The four scores must add to 4×85=3404 \times 85 = 3404×85=340. Subtract the three known scores.
340−(78+85+90)=340−253=87340 - (78 + 85 + 90) = 340 - 253 = 87340−(78+85+90)=340−253=87
Find the median of 313131, 272727, 353535, 292929, 333333.
Correct answer: C
Sort the values: 27,29,31,33,3527, 29, 31, 33, 3527,29,31,33,35. With five values the third one is the middle.
median=31\text{median} = 31median=31
Which measure of center is least affected by a single outlier?
Correct answer: A
The mean uses the size of every value, so an outlier pulls it. The median depends only on the middle position, so one extreme value barely moves it.
least affected by an outlier=the median\text{least affected by an outlier} = \text{the median}least affected by an outlier=the median
Find the median of 888, 121212, 444, 202020, 161616, 101010.
Sort the values: 4,8,10,12,16,204, 8, 10, 12, 16, 204,8,10,12,16,20. With an even count, average the two middle values, 101010 and 121212.
median=10+122=11\text{median} = \frac{10 + 12}{2} = 11median=210+12=11
For the data 666, 666, 777, 888, 888, 888, 999, what are the mode and the median?
The value 888 appears three times, more than any other, so it is the mode. The data is sorted and has seven values, so the median is the fourth value.
mode=8,median=8\text{mode} = 8, \qquad \text{median} = 8mode=8,median=8
If every value in a data set decreases by 333, what happens to the mean?
Subtracting 333 from each of the nnn values removes 3n3n3n from the sum. Dividing that by nnn lowers the mean by 333.
new mean=old sum−3nn=old mean−3\text{new mean} = \frac{\text{old sum} - 3n}{n} = \text{old mean} - 3new mean=nold sum−3n=old mean−3
A list of 777 numbers has a mean of 666. You add one more number that is equal to 666. What is the new mean?
Correct answer: D
The seven numbers add to 424242. Adding a value equal to the mean gives a sum of 484848 over 888 values.
new mean=42+68=488=6\text{new mean} = \frac{42 + 6}{8} = \frac{48}{8} = 6new mean=842+6=848=6
Adding a value equal to the mean never changes the mean.
Find the range of −6-6−6, −1-1−1, 444, −9-9−9, 333.
The largest value is 444 and the smallest is −9-9−9. Subtract, being careful with the negative.
range=4−(−9)=4+9=13\text{range} = 4 - (-9) = 4 + 9 = 13range=4−(−9)=4+9=13
Which statement about the range is true?
The range is the largest value minus the smallest, so it describes how stretched out the data is rather than where the center sits.
range=max−min ⇒ a measure of spread\text{range} = \text{max} - \text{min} \;\Rightarrow\; \text{a measure of spread}range=max−min⇒a measure of spread
How many modes does the data set 777, 777, 888, 999, 101010, 101010 have?
Count each value: 777 appears twice and 101010 appears twice, while 888 and 999 appear once each. Two values tie for the most appearances.
modes=7 and 10\text{modes} = 7 \text{ and } 10modes=7 and 10
For the data 121212, 151515, 181818, 212121, 242424, which statement is true?
The sum is 909090, so the mean is 905=18\frac{90}{5} = 18590=18. The values are sorted and the middle one is also 181818.
mean=18,median=18\text{mean} = 18, \qquad \text{median} = 18mean=18,median=18
The test scores 606060, 707070, 808080, 909090, 100100100 have the lowest score, 606060, replaced by 000. Which changes: the mean, the median, or both?
The middle value stays 808080, so the median does not change. The sum drops from 400400400 to 340340340, so the mean falls.
old mean=4005=80,new mean=3405=68\text{old mean} = \frac{400}{5} = 80, \qquad \text{new mean} = \frac{340}{5} = 68old mean=5400=80,new mean=5340=68
Only the mean changes, because the replaced value is not in the middle.
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