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12 Jun 2020, 02:22
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The disqualification of Sauer would decrease the standard deviation more than any other single disqualification.: True
The disqualification of Lasek would decrease the standard deviation more than any other single disqualification.: False
If Fournier and Sauer were both disqualified and no other changes were made, the standard deviation would decrease.: True
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Ten people participated in the first round of a competition, after which their scores were analyzed to determine which people would compete in the second round. The table shows the participants’ first-round scores, the difference between each participant’s score and the average (arithmetic mean) score of all participants, and the square of that difference.
Before the second round of competition, one or more of the listed participants were disqualified. A new mean and new standard deviation will be calculated from only those players who were not disqualified.
For each of the following statements, select True if the statement is true based on the information provided. Otherwise, select False.
Before the second round of competition, one or more of the listed participants were disqualified. A new mean and new standard deviation will be calculated from only those players who were not disqualified.
| Participant Name | Score | Difference from Mean | Square of Difference from Mean |
|---|---|---|---|
| Arceneaux | 56 | —9 | 81 |
| Cabrera | 89 | 24 | 576 |
| Costantino | 34 | —31 | 961 |
| Fournier | 24 | —41 | 1681 |
| Hough | 109 | 44 | 1936 |
| Keffala | 56 | —9 | 81 |
| Lasek | 120 | 55 | 3025 |
| Sauer | 6 | —59 | 3481 |
| Tsakiris | 78 | 13 | 169 |
| Willems | 78 | 13 | 169 |
For each of the following statements, select True if the statement is true based on the information provided. Otherwise, select False.
| True | False | |
| The disqualification of Sauer would decrease the standard deviation more than any other single disqualification. | ||
| The disqualification of Lasek would decrease the standard deviation more than any other single disqualification. | ||
| If Fournier and Sauer were both disqualified and no other changes were made, the standard deviation would decrease. |
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Official Answer
The disqualification of Sauer would decrease the standard deviation more than any other single disqualification.: True
The disqualification of Lasek would decrease the standard deviation more than any other single disqualification.: False
If Fournier and Sauer were both disqualified and no other changes were made, the standard deviation would decrease.: True
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12 Jun 2020, 03:45
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1. The disqualification of Sauer would decrease the standard deviation more than any other single disqualification.
Sauer's score = 6, Diff w.r.t mean = -59(lowest in all participants).
=> Mean = 59 + 6 = 65.
Now standard deviation shows how much variation there is from the mean in a set.
Low SD => values are very close in the set,
High SD => values are spread across the set.
Since, the difference w.r.t mean is lowest for Sauer, and the score is also lowest for Sauer.
His disqualification, would decrease the standard deviation more than anyone else's, since it would increase the mean, reduce the difference w.r.t mean and thus lower the SD.
Answer True.
2. The disqualification of Lasek would decrease the standard deviation more than any other single disqualification.
False since disqualification of Sauer would decrease the standard deviation more than any other single disqualification.
Also, if Lasek is removed, => Mean change = (650 - 120)/9 = 530/9 = 58.8.
=> the diff. from the mean would reduce, => the SD would increase.
Answer False.
If Fournier and Sauer were both disqualified and no other changes were made, the standard deviation would decrease.
Fournier's score = 24, Sauer's score = 6.
New mean = (650 - 30)/8 => 620/8 => new mean = 77.5.
Now since these 2 participants have the lowest scores, if they are disqualified, the difference from the mean would improve, which will result in lowering of SD.
Answer True.
Sauer's score = 6, Diff w.r.t mean = -59(lowest in all participants).
=> Mean = 59 + 6 = 65.
Now standard deviation shows how much variation there is from the mean in a set.
Low SD => values are very close in the set,
High SD => values are spread across the set.
Since, the difference w.r.t mean is lowest for Sauer, and the score is also lowest for Sauer.
His disqualification, would decrease the standard deviation more than anyone else's, since it would increase the mean, reduce the difference w.r.t mean and thus lower the SD.
Answer True.
2. The disqualification of Lasek would decrease the standard deviation more than any other single disqualification.
False since disqualification of Sauer would decrease the standard deviation more than any other single disqualification.
Also, if Lasek is removed, => Mean change = (650 - 120)/9 = 530/9 = 58.8.
=> the diff. from the mean would reduce, => the SD would increase.
Answer False.
If Fournier and Sauer were both disqualified and no other changes were made, the standard deviation would decrease.
Fournier's score = 24, Sauer's score = 6.
New mean = (650 - 30)/8 => 620/8 => new mean = 77.5.
Now since these 2 participants have the lowest scores, if they are disqualified, the difference from the mean would improve, which will result in lowering of SD.
Answer True.
05 Apr 2021, 03:17
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Official Explanation
RO1
The standard deviation is found by averaging the squared differences from the mean and then taking the square root of that average. That average, and thus the standard deviation, would thus decrease the most by removing the largest number in the Square of Difference from Mean column. Sorting the table by Square of Difference from Mean shows that the largest value in that column is for the score of Sauer.
The correct answer is True.
RO2
As discussed in the analysis of RO1, the standard deviation would decrease the most by removing the score with the largest entry in the Square of Difference from Mean column. That score is held by Sauer, not Lasek.
The correct answer is False.
RO3
The standard deviation would decrease if the average of the squared differences from the mean decreased. The average after the first round, found by taking the sum of the values in that column (12,160) and dividing by the number of participants (10), is 1,216. The scores of both Fournier and Sauer have squared differences from the mean that exceed this average (1,681 and 3,481, respectively). Therefore, removing those values would decrease both the average and the standard deviation.
The correct answer is True.
RO1
The standard deviation is found by averaging the squared differences from the mean and then taking the square root of that average. That average, and thus the standard deviation, would thus decrease the most by removing the largest number in the Square of Difference from Mean column. Sorting the table by Square of Difference from Mean shows that the largest value in that column is for the score of Sauer.
The correct answer is True.
RO2
As discussed in the analysis of RO1, the standard deviation would decrease the most by removing the score with the largest entry in the Square of Difference from Mean column. That score is held by Sauer, not Lasek.
The correct answer is False.
RO3
The standard deviation would decrease if the average of the squared differences from the mean decreased. The average after the first round, found by taking the sum of the values in that column (12,160) and dividing by the number of participants (10), is 1,216. The scores of both Fournier and Sauer have squared differences from the mean that exceed this average (1,681 and 3,481, respectively). Therefore, removing those values would decrease both the average and the standard deviation.
The correct answer is True.
