6 exercises · 6 automatically checked · Untimed
6 questions · Shuffled order · Untimed practice
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An integer observation belongs to the histogram bin [−3, 2). What is its greatest possible value?
A: −3 is the included minimum, not the maximum.
B — correct: 1 is the largest integer strictly below 2.
C: The upper endpoint is excluded.
D: 1.9 is below 2 but is not an integer.
A data set has three integers in [2, 5) and two in [5, 8). What is its greatest possible mean?
A: This uses the excluded upper endpoints 5 and 8.
B: 4 is only the maximum for the first bin.
C — correct: Use 4, 4, 4, 7, 7. Their sum is 26 and their mean is 26/5 = 5.2; all five values are allowed.
D: This exceeds the attainable maximum, since even every individual upper feasible value gives only 5.2.
The histograms describe independent integer data sets. Find the smallest possible value of mean A minus mean B.
A: This uses B’s excluded upper endpoints, giving a mean of 5.25 instead of 4.25.
B: 25 is the difference of the endpoint sums 42 and 17; unequal counts require separate means.
C: There is no paired shift established by these different frequencies.
D — correct: A can be 6, 6, 6, 6, 9, 9, with mean 7. B can be 2, 5, 5, 5, with mean 4.25. Their difference 2.75 is attained and minimal.
Four integer observations are grouped as follows: three in [10, 15) and one in [15, 20). What is the smallest possible mean?
Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.
Use the included lower endpoints: three observations of 10 and one of 15.
Minimum mean = = = 11.25.
The list 10, 10, 10, 15 proves attainability. Using midpoints estimates a different mean; using 14 and 19 maximizes rather than minimizes it. Enter 45/4 or 11.25.
A data set has one integer in [2, 6), four in [6, 10), and one in [10, 14). A new value of 0 is added. Which statement must be true?
A — correct: Zero is below every old value and hence the old mean, so the mean decreases. For 3, 7, 7, 7, 7, 11, adding 0 leaves the median at 7.
B: The displayed grouping allows equal central values, as the counterexample in A shows.
C: The mean must decrease, while equal central values can keep the median unchanged.
D: The mean cannot remain unchanged when a value strictly below it is added.
Four observations may be any real numbers in [1, 4); they need not be integers. Which statement about their greatest possible mean is correct?
A: Real values such as four copies of 3.9 give a larger mean.
B: Every value is strictly below 4, so their mean is also below 4.
C — correct: Choosing four copies of 4 minus any small positive amount gives a mean as close to 4 as desired, but never equal to 4.
D: 2.5 uses the midpoint; the interval does not fix the individual values.
Review Exact bounds from grouped data