Statistics Demystified, 2nd edition

Stan Gibilisco

Explanations for Quiz Answers in Chapter 4

1. In this context, the range equals the highest actual student score minus the lowest actual student score. That's 18 - 2, or 16. The answer is C.

2. The third quartile point in a set of student test scores is the largest score in the set of the "worst" 3/4 (75%) or fewer scores. In this case we have a total of 80 scores (the class has 80 students), and 60 equals 3/4 of 80. The third quartile point therefore corresponds to the highest score representing the "worst" 60 or fewer papers. The correct choice is D.

3. We've been informed that the mean score ľ equals 66, while the standard of deviation σ equals 13.2. The coefficient of variation CV equals the standard of deviation divided by the mean:

CV = σ / ľ
= 13.2 / 66
= 0.20

The correct choice is D.

4. In a statistical distribution, a particular element's Z score equals the number of standard deviations by which that element differs from the mean (either positively or negatively). This question asks us to determine the Z score for an entire test; that's the wrong context! Moreover, we don't know the standard deviation or the mean. The correct choice is D, "None of the above," because we can't define a Z score in this situation.

5. In the graph described, the "half-pie" slice accounts for 50% of the families in Happytown earning incomes within a certain range. However, you can't possibly figure out that range, because the graph doesn't give you any numerical data! Although any one of the three choices A, B, or C might represent the truth, you lack the information to say which one it is (or whether that "half-pie" slice represents some "oddball" range such as the proportion of people whose incomes put them in the lowest tax bracket). The correct choice is D, "None of the above."

6. In the situation that Table 4-11 breaks down for us, the second decile point portrays the "worst" 2/10 (20%) or fewer test scores. We have 140 student test papers in total; 20% of that figure equals 28 papers. We don't see the number 28 anywhere in the "Cumulative Absolute Frequency" column of Table 4-11, but we can see that the "28th worst" paper must be between scores of 4 and 5. The second decile point lies in the transition or "fuzzy zone" between scores of 4 and 5, so the correct choice is B.

7. At least one student got every possible score. The highest paper scored 10, and the lowest paper scored 0. The range, in this context, equals the highest actual score minus the lowest actual score; that's 10 - 0, or 0. The answer is C.

8. In the situation shown by Table 4-11, the 10th percentile point portrays the "worst" 10/100 (10%) or fewer test scores. Of the 140 student test papers, 10% represents 14 of them. That number doesn't appear in the "Cumulative Absolute Frequency" column, but we can see that the "14th worst" paper must be between scores of 3 and 4. The 10th percentile point lies in the transition region between scores of 3 and 4. The answer is A.

9. The second highest 10% corresponds to the range of scores bounded at the bottom by the eighth decile and at the top by the ninth decile. The eighth decile is the highest possible boundary point at the top of the set of the "worst 80%" or fewer papers. In Table 4-11, that point corresponds to a cumulative absolute frequency of 112 (which is 80% of 140). The ninth decile constitutes the highest possible boundary point at the top of the set of the "worst 90%" or fewer papers. That point corresponds to a cumulative absolute frequency of 126 (which is 90% of 140). Mary M. got a score of 8, so the table places her at the cumulative absolute frequency level of 123. Because 123 lies between 112 and 126, we know that Mary M. scored in the second highest 10% of the class. The correct choice is B.

10. By definition, the interquartile range equals the value of the third quartile point minus the value of the first quartile point. It's the "central half" of the data in the distribution. The third quartile point is the highest possible boundary representing the "worst" 3/4 (75%) or fewer papers, in this case the "worst" 105 or fewer (because 105 equals 75% of 140). That cumulative-frequency number lies between scores of 6 and 7 according to the table, so its score theoretically equals 6.5. The first quartile point is the highest possible boundary representing the "worst" 1/4 (25%) or fewer papers, in this case the "worst" 35 or fewer (because 35 is 25% of 140). That cumulative-frequency number lies between scores of 4 and 5 in the table, so its score theoretically equals 4.5. The interquartile range is therefore 6.5 - 4.5, or 2. The answer is A.