Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 A B C D

Solutions for Session 9, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6

Problem C1

 a. A good estimate might be the median of the 100 estimates, which is in position (100 + 1)/2 = 50.5. This means that the median is the average of the 50th and 51st values in the ordered list. Both the 50th and 51st values are 500, so 500 penguins is a good estimate, based on the median. b. It seems very likely that the actual number is between 360 and 620, since all 100 estimates fall in this range. A tighter range is 450 to 550, which includes 69 of the 100 estimates.

Problem C2

 a. The best estimate is 500, which is exactly right. Our sampling found this estimate eight of 100 times. b. The six worst estimates are 360, 360, 390, 610, 610, and 620. These are the only six estimates that are more than 100 penguins away from the actual value. c. These are the estimates between 450 and 550 (inclusive); 69% (69/100) of the estimates are within this range. d. These are the estimates between 400 and 600 (inclusive); since only six estimates are more than 100 penguins away, 94% (94/100) of the estimates are within this range.

Problem C3

 a. These are the estimates between 425 and 575 penguins (inclusive); the proportion is 84/100, since 84 of the 100 estimates are within this range. b. The proportion is 16/100 (obtained as 1 - 84/100).

Problem C4

Here is the completed table:

Interval (inclusive)

Proportion of Estimates in Interval

Proportion of Estimates Not in Interval

 350-650 100/100 0/100 375-625 98/100 2/100 400-600 94/100 6/100 425-575 84/100 16/100 450-550 69/100 31/100 475-525 37/100 63/100

Problem C5

 a. When the proportion of estimates in an interval is high, it is a strong suggestion that the actual population value lies somewhere in that range. b. A small interval gives greater precision to the estimates. If we can say that the actual value lies between 475 and 525, it is more meaningful than saying that the actual value lies, say, between 400 and 600. c. As the interval range decreases, the proportion of estimates in that interval decreases. Thus, there is an important tradeoff: A wide interval will contain more estimates but will be less meaningful, whereas a small interval will be more meaningful but will contain fewer estimates.

Problem C6

 a. The expected probability is 0.84, or 84%, since 84 of the 100 estimates fall in this interval. b. The probability is 37%, since 37 of the estimates fall in the smallest interval (475 to 525). c. It is very likely -- 94% of the estimates fall in the interval within 100 penguins of the actual total (400 to 600).

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