Self-check: 360-300, Quantitative Methods in Social Science
One question per chapter, 8 in all, taken from the course's corrected exercises. Answer the ones you can do and skip those from a chapter you have not seen yet: each answer is checked right away, and the solution opens after.
With your free account, each answer is kept and your dashboard files each chapter: passed the check, or to work on. These are the same questions as on the set pages: an answer given here counts there.
Chapter 1 · Concepts, variables and measurement scales
An instrument can be wrong in the same way every time. The target below separates the two qualities we demand of a measurement: aiming true, and aiming tight.
b) A bathroom scale systematically reads 2.0 kg too high, with a repeatability of 0.1 kg. Where do you place it among the four cases? What is it worth for tracking weight loss?
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Answer
b)Case 2, reliable and invalid; the measured difference is −2.5 kg, exact, because the bias cancels out
b) This scale is case 2: very RELIABLE, with a repeatability of 0.1 kg, and INVALID for measuring an absolute weight, because of the constant 2.0 kg bias. For tracking weight loss, however, it is perfectly usable, because the quantity of interest is a DIFFERENCE: (p2+2)−(p1+2)=p2−p1, the constant bias cancels out. That is the practical lesson of the exercise: the quality required depends on the question asked, and a biased but stable instrument remains good for comparing, bad for locating.
Size is computed from the precision wanted, never from the size of the population. It is the most counterintuitive result of the chapter, and the most useful.
We want to estimate a proportion at the 95% level, for which z=1.96, with the formula n=E2z2p(1−p).
a) Compute n for a margin of 3 percentage points, then of 2 points, taking p=0.5. Comment on the ratio between the two.
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Answer
a)n≈1,067 for 3 points, 2,401 for 2 points; the ratio is (23)2=2.25
a) For E=0.03: n=0.0321.962×0.25=0.00090.9604≈1,067. For E=0.02: n=0.00040.9604≈2,401. The ratio is 10672401≈2.25=(23)2: dividing the margin by 1.5 multiplies the size by 1.5 squared. The margin decreases as n1, so the size grows as the square of the precision demanded, and that is what makes very precise surveys prohibitively expensive.
The histogram is the graph of the continuous variable, and the only one whose bars touch. The polygon that crowns it serves another purpose: comparing two distributions on the same drawing.
The weekly study hours of 86 students were grouped into six classes of 5 hours.
b) Why do the bars of a histogram touch, while those of a bar chart do not?
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Answer
b)Continuous: the classes follow each other edge to edge. Discrete or qualitative: emptiness between the categories
b) Because the variable is CONTINUOUS: between 9.9 and 10.1 hours there is no gap, the value 10 exists, and the classes follow each other edge to edge, covering the whole interval. A bar chart represents a discrete or qualitative variable, where the categories are separated by emptiness: between 2 and 3 courses failed, no value exists, and leaving a space says so visually. The difference in the drawing therefore reflects a difference in nature, and it is exactly the kind of detail on which a marker takes off points.
Chapter 4 · Z scores, percentile ranks and the R score
The Z score replaces each value by its distance from the mean, counted in standard deviations: z=sx−x. The operation looks harmless, yet it completely changes what the number says.
Five exam results: 70, 74, 78, 82, 86.
c) Everyone is given 5 extra points. What happens to the Z scores? And if all the marks are multiplied by 1.2?
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Answer
c)Unchanged in both cases: the Z score is invariant under any increasing affine transformation
c) Adding 5 points to everyone shifts the mean by 5 and leaves the standard deviation unchanged: each numerator xi+5−(x+5) gives back xi−x, so the Z scores are UNCHANGED. Multiplying by 1.2 multiplies both the mean and the standard deviation by 1.2: the numerator and the denominator are both multiplied by 1.2, and the Z scores are still UNCHANGED. This is why standardization is the right tool for comparing teachers who do not mark on the same scale: it is insensitive to any increasing affine transformation.
Chapter 5 · Index numbers, rates and constant dollars
A rate is a quotient, and what matters is the denominator. The unemployment rate is the most disputed example, because the choice of who counts as unemployed changes the result by several points.
c) The number of jobs stays the same but 12,000 unemployed people leave the labour market. What happens to the unemployment rate?
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Answer
c)404,00020,000≈5.0%: 2.7 points lower without a single job created
c) The labour force falls to 416,000−12,000=404,000 and the unemployed to 20,000. The rate becomes 404,00020,000≈0.0495, that is 5.0% instead of 7.7%. The rate has therefore DROPPED by 2.7 points although no job has been created: people simply stopped looking. This is the exact mechanism of discouragement, and it operates strongly at the end of a recession.
Chapter 6 · Cross-tabulations, chi-square and association
The chi-square adds up the squared differences, each one scaled by what was expected. It then yields a decision and, above all, if you read it cell by cell, it tells you WHERE the relationship is.
χ2=∑E(O−E)2, with df=(r−1)(c−1). Critical values at 5%: 3.841 for 1 degree of freedom, 5.991 for 2, 7.815 for 3, 9.488 for 4.
e) Does rejecting the null hypothesis prove that the relationship is strong? What else must be computed?
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Answer
e)No: rejection says nothing about strength, which is read on Cramér's V
e) No. Rejection says that the observed gap is too large to be attributed to sampling chance alone; it says nothing about the STRENGTH of the relationship. The proof is that, for an identical structure, the chi-square grows in proportion to the sample size: any gap, however tiny, becomes significant with enough respondents. You must therefore also compute a normalized measure of ASSOCIATION, Cramér's V or the contingency coefficient, which does not depend on n.
A poll gives 42% to candidate A and 38% to candidate B, with a margin of 3 points. A four-point lead, a three-point margin: is the lead established? The answer is no, and the reason deserves a computation.
The figure below represents the situation.
a) Write the two confidence intervals, each with a margin of 3.1 points. Do they overlap?
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Answer
a)[38.9,45.1] and [34.9,41.1]: they overlap over 2.2 points
a) Candidate A: [38.9,45.1]. Candidate B: [34.9,41.1]. They overlap over the whole zone from 38.9 to 41.1, a common width of 2.2 points. A hurried reader would conclude that nothing can be said.
What a study is entitled to conclude depends neither on its size nor on the sophistication of its calculations, but on its DESIGN. The scale below is the one a reader must have in mind before reading a single result.
b) For each one, say whether it can establish the TEMPORAL PRECEDENCE of the cause over the effect.
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Answer
b)Temporal precedence: sometimes for the case study, NO for cross-sectional, YES for longitudinal and experimental, partial for quasi-experimental
b) Case study: sometimes, if the chronological reconstruction is solid, but on a single unit. Cross-sectional: NO, that is its fundamental weakness, everything is measured at the same time and nothing says what came first. Longitudinal: YES, that is precisely its contribution, repeated measurement establishes the time order. Quasi-experimental: yes for the treatment, whose date is known, no for the preexisting differences between the groups. Experimental: YES, the treatment is given after the groups are formed, on a date known to the researcher.
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