The "Must-Pass" Quiz for 2003: Partial Answer Key

AP Statistics / Mr. Hansen
5/5/2003

Name: _________________________

The “Must-Pass” Quiz for 2003: Partial Answer Key

1.*	A number computed from data. [You should provide examples.]

2.*	A number that describes a population. [You should provide examples.]

3.*	An “adjustable constant” that defines the nature of a mathematical model, much as a tuning knob or volume slider adjusts the output of a television or radio.

4.	Uniform: min and max [also need to know whether distrib. is discrete or continuous] Normal: m and s Binomial: n and p Geometric: p t: df c²: df

5.	Uniform: flat line in relative frequency histogram Normal: classic continuous bell-shaped curve, satisfies 68-95-99.7 rule Binomial: discrete (“stairsteppy”); skew right if p < .5, skew left if p > .5, symmetric if p = q = .5 Geometric: discrete (“stairsteppy”), always skew right t: continuous, bell-shaped; virtually normal for large df, except with more “flab” in the tails c²: continuous, always skew right

6.	Range is a single number for the spread of values in a column of data: range = max – min. People who say things like “the range is from 28 to 75” are misusing the term in its statistical sense.

7.	IQR (interquartile range) = Q₃ – Q₁. Use STAT CALC 1 to get 5-number summary, then VARS 5 PTS 9 – VARS 5 PTS 7. You could write a program to do this if you wished.

8. (a)	Easiest way is to make modified boxplot, then TRACE to see the points (use arrow keys). Outliers are more than 1.5IQR below Q₁ or more than 1.5IQR above Q₃.

(b)	No rule of thumb—just judge visually. Outliers have “large” residuals.

9.*	Explanatory, response.

10.	Mean squared error = pop. variance (mean squared deviation from the mean). Sample variance is different, since denom. is n – 1 instead of n.

11.	Pop. s.d. (s) and sample s.d. (s) are measures of data dispersion (“spread”). Use STAT CALC 1 to compute, never the formula on AP formula sheet. Technically, s equals the square root of MSE (square root of pop. variance), and s equals the square root of sample variance.

12.	In a normal distribution (required), the distribution curve is bell-shaped, satisfies the 68-95-99.7 rule, and has inflection points at ±1s.

13.	Lack of symmetry. Right skewness means the central hump dribbles out to the right, forcing mean > median, since mean is less resistant to extreme values. Right skewness is the opposite, forcing mean < median. Easy ways to detect skewness involve looking at histogram, boxplot, or stemplot to see where the tail is longer. If you use NQP, trace dots from left to right; if they bend to left, plot shows left skewness, but if they bend to right, plot shows right skewness.

14.	Easiest way is look for a pattern that is not straight in NQP. If you are a glutton for punishment (as on #4 from the Chap. 13-14 free response), you can use c² g.o.f. to test for departures from expected bin counts. There are also several standard “canned” tests that are beyond the scope of AP Statistics.

15.*	Linear.

16.	Slope, since it estimates how many response units will increase (or decrease) for each additional explanatory unit. Intercept is less crucial, even meaningless in some contexts.

17.	Linear correlation coefficient. Signed strength of linear pattern (–1 = pure negative linear association, 0 = no linear association, +1 = pure positive linear association.) Use STAT CALC 8 and make sure your Diagnostics are on (2nd CATALOG DiagnosticOn).

18.	Coefficient of determination. Tells what portion of the variation in one variable can be explained by variation in the other. If r = .8, then 64% of the variation in y (or x) can be explained by variation in x (or y).

19.	No; yes.

20.	No; yes.

21.	STAT CALC 8, or with formulas 6 and 8 on first page of AP formula sheet. (Never use formula 5.)

22.	[See LSRL Top Ten.]

23.	Resid. = y – yhat (i.e., actual y – predicted y). Resid. plot is scatterplot with RESID on y-axis and either the x or y variable on the x-axis. (It doesn’t matter, since x and y are linearly related.) In beginning statistics courses, we usually make resid. plot with x on the x-axis and RESID on the y-axis, but there was at least one AP exam that had y values on the x-axis of the resid. plot. Don’t let that bother you.

24.	[See LSRL Top Ten.]

25.	Regression outlier and influential observation are not synonyms. A point can be a regression outlier (large residual), but if it is near the center of the x values, it is usually not influential. Similarly, a point can be influential (large effect on slope or r if removed) but have only a small residual, meaning the point is not an outlier. It is also possible for a point to be both influential and an outlier.

26.	b₀ = value of response if explanatory variable (x value) is set to 0 b₁ = estimate of how many response units will increase (or decrease) for each additional explanatory unit For example, suppose that a clinical trial of a diet pill shows that the mean weight change after a year is 2 – 3x lbs., where x = daily dosage (# of pills). Then b₀ = 2, since a person taking 0 pills can expect to gain 2 lbs. in a year, and b₁ = –3, since each additional pill in the daily dosage is associated with a weight of about 3 lbs. less after a year.

27.	Random variable (discrete or continuous). [You should provide examples.]

28.	r.v., Sp_ix_i, mean, expected value

29.	r.v., variance, Var(X), s²_X, square root, Var(X), s_X

30.	sum, sum, means; yes; mean of difference equals difference of means

31.	sum, sum, variances; true only for independent r.v.’s; variance of difference (assuming indep. r.v.’s) equals sum of variances Other consequences: s.d. of sum = square root of sum of variances (similar to Pythagorean Theorem), s.d. of difference = square root of sum of variances (same comment). Both are true only if the r.v.’s are independent.

32.	scalar (i.e., a constant), scalar, s_X; yes

33.	r: no change m: affected by both translation and dilation (fancy way of saying that m_new = lin. fcn. of m_old) s: affected by dilation (i.e., multiplication by scalar) but not by translation (shift left or right) IQR: affected by dilation but not by translation range: affected by dilation but not by translation

34.	Standardized (dimensionless) representation of a data point, in s.d.’s. Can always be computed, even if data set is non-normal. Use formula z = (x – m)/s. Tells how many s.d.’s a data value is above or below the mean.

35.	events, mutually exclusive; independence; no; independence of A and B means P(A\|B) = P(A), which is not at all the same as P(A Ç B) = 0

36.*	The aspect of probability that we care most about is sampling distributions. If we understand the sampling distribution of a statistic, we can determine how statistically significant a result is. Without this, we would never know whether experiments or clinical trials of new drugs were showing anything of value or were merely “flukes.”

37.	Sampling distribution of xbar or diff. of means: Follows z if s is known (rare), otherwise t. Sampling distribution of phat: Really binomial, but almost normal if pop. is large, np ³ 10, and nq ³ 10. Sampling distribution of difference of proportions: Almost normal if pops. are large, n₁p₁ ³ 5, n₁q₁ ³ 5, n₂p₂ ³ 5, n₂q₂ ³ 5. Sampling distrib. of S(obs. – exp.)²/exp.: Follows c², with df given either by (# of bins – 1) for g.o.f., or by (rows – 1)(cols. – 1) for 2-way tables.

38.	s.e.; no idea; yes

39.*	Law of large numbers. CORRECT: As n ® ¥, phat approaches p. (Sometimes stated as “xbar approaches m as n ® ¥.) WRONG: If phat < p, then the proportion of successes will start to increase until we “catch up.” (Or, if phat > p, the proportion of successes will start to decrease until we are “back down to the correct value.”) These are both wrong, because what really happens is that the effect of any finite collection of observations becomes diluted as n ® ¥. A coin has no memory, no desire to set things right, and no ability to iron out past discrepancies. Nevertheless, the proportion of heads—even if the coin is biased—will, over time, approach whatever the true probability is.

40.	Central limit theorem. CORRECT: Consider any population, not necessarily normal, having finite s. As n ® ¥, the sampling distribution of xbar approaches N(m, s/Ön). WRONG: “Everything is normal.” (Not true: Sampling distributions of s are certainly not normal. Geometric and c² distributions are certainly not normal.) “Any sampling distribution of xbar is normal.” (Not true: Sampling distributions of xbar approximately follow a t distribution if s is unknown.) “Sampling distribution of xbar is not normal unless n is large.” (False. Sampling distribution of sample mean is normal if pop. is normal with known s, regardless of sample size.)

41.	P-value, test; principles of good experimental design; [add your personal description]

42-52.	[Research on your own, please.]

53.	Two-tailed, since if the experiment goes the wrong way (as sometimes occurs in science), there will still be the possibility of making an inference. All decisions regarding methodology are supposed to be made before any data-gathering occurs. (Otherwise, people could say that the methodology was tailored toward achieving a low P-value. In theory, the experiment should be repeatable, so that anyone following the same methodology would likely reach a similar conclusion.) The one-tailed/two-tailed decision should be based on the research question posed. If the researcher is wondering whether there is “a difference,” direction unspecified, then plan for a two-tailed test. If the researcher is wondering whether treatment X increases hair strength, decreases yellowness of teeth, or whatever, then plan for a one-tailed test.

54.	It is possible to write a true sentence using the words probability and confidence interval. However, it is also very easy to make an error along the way. That is why it is much better to say, “We are 95% confident that the true proportion of voters favoring candidate Smedley is between 48% and 54%,” not anything involving probability. Probability is a technical term meaning long-run relative frequency, and it cannot be haphazardly misused in the way laypeople misuse it. It would be correct to say, “If we repeatedly generated confidence intervals with samples of this size and with m.o.e. of 3%, then the probability that a future confidence interval will bracket the true proportion of voters favoring candidate Smedley is 95%; that is, 95% of the confidence intervals generated by this process will bracket the true value.” However, you cannot make a probability statement about a confidence interval once it has been generated, because then you are not making a statement about the process (which is legitimate), but rather about this one-shot confidence interval. There is no “long run” in a one-shot confidence interval!

55.	We cannot prove H₀. All we can do is judge whether the evidence against it is “sufficient to reject” or “insufficient to reject.”

56.	We can sometimes gather overwhelming evidence that H₀ can be rejected in favor of H_a. In the real world, even in a court of law, that is good enough. (Of course, in the world of mathematics, that is not considered a proof—one of the reasons that mathematicians and statisticians do not consider themselves to be equivalent.)

57.*	[You’d better know this by now!]

58.*	inferential, use statistics to estimate parameters

59.	[I think everyone can do this.]

60.	Always use the first one, never the second.

61.	The first one (unequal proportions) is for a 2-prop. z confidence interval, and the second one is usually for a 2-prop. z test. The only exception would be if you had H₀ stating something other than p₁ = p₂, but that is rare. In the second formula, you have to know how to estimate p: Take total # of successes divided by total # of subjects.

62.	False: If there are matched pairs, you really have only one sample (namely, a column of differences).

63.	Systematic departure from randomness, i.e., a methodology that produces samples that are systematically different from the population in a way that causes a parameter to be systematically underestimated or overestimated. An SRS is not biased; although an SRS often fails to match the population, the differences are random differences, not systematic differences. Systematic means that there are flaws in the methodology. Common types of bias include undercoverage, overcoverage, response bias (a.k.a. lying), nonresponse bias, voluntary response bias, hidden bias, and wording of the question.

64.	xbar is an unbiased estimator of m; i.e., E(xbar) = m_xbar = m phat is an unbiased estimator of p; i.e., E(phat) = m_phat = p

65.	[I hope you have thought about this. This is a personal matter, but what I do is first to decide whether there are proportions involved or not. Then, do we have 1 sample, matched pairs (also 1 sample), or 2 real samples? Or is this a c² problem? And if so, are we comparing against fixed proportions (g.o.f.) or looking for differences across a 2-way table?]

66-74.	[See TI-83 STAT TESTS Summary.]

75.	“There is strong evidence that ...” It is a good idea to list the test statistic, n or df, and the P-value in parentheses. Be sure to phrase the conclusion in the context of the problem.

76.	“There is insufficient evidence that ...” It is a good idea to list the test statistic, n or df, and the P-value in parentheses. Be sure to phrase the conclusion in the context of the problem.

77.	“We are XX% confident that the true ... is between YY and ZZ.” Be sure to phrase the “...” in the context of the problem, e.g., “true mean boiling point,” “true difference in voter preference proportions,” “true mean improvement in test scores,” etc.

78.	Compute C.I. using TI-83. Then punch upper–lower, i.e., VARS 5 TEST I – VARS 5 TEST H, divide result by 2 and STO into M (for m.o.e.). Your can then write your C.I. as est. ± M. Depending on the problem, “est.” will be xbar, phat, xbar₁ – xbar₂, or phat₂ – phat₂.

79.	[See AP formula sheet.]

80.	Since t = (b₁ – 0)/s.e. = b₁/s_b₁ in the LSRL t-test, s_b₁ = b₁/t.

81.*	convenience, anecdotal; voluntary response bias

82.*	No. [We talked about this on the very first day of class and on numerous occasions since then. Please provide an illustrative example.]

83.	No; q; yes.

84.	Simple random sample; a sample in which every possible subset is equally likely to be selected.

85.	SRS, since bias can invalidate the results quite easily. Normality of population is not an issue in large samples (courtesy of CLT), since normality of the sampling distribution rescues us.

86.	Marginal probabilities = fractions involving row or column totals divided by grand total. Conditional probabilities = fractions involving individuals cells divided by a row or column total. Both are usually concerned with categorical data in 2-way tables.

87.*	Just because an effect is not plausibly caused by chance alone does not mean that it is large enough to be of any real-world significance.

88.*	[I think everyone knows this. In fact, you probably knew it before you ever signed up for the course.]

89.*	Only a controlled experiment is considered convincing. In situations (e.g., smoking in humans) where it is not ethical to run a controlled experiment, various types of observational and correlative studies can suggest, but not prove, a cause-and-effect link.

90.*	[Everyone probably knows this. Remember to discuss placebo effect and hidden bias.]

91.	Yes; perhaps many new employees have been hired.

92.	Yes; the relative mix of employee categories could be a lurking variable. Perhaps there are now proportionally more employees in the higher-paid job categories, so that the weighted average salary has increased even while each category has had cuts in mean salaries. This would be an example of Simpson’s Paradox.

93.*	Using deceptive (“gee-whiz”) graphs, changing the subject, confusing correlation with causation, using inappropriate averages (e.g., mean with highly skewed distributions), citing anecdotal data, using biased samples, concealing the wording of a survey question, computing absurd precision with qualitative data (e.g., “74% more beautiful skin!”), etc., etc.

94.*	Who says so? How do they know? Did somebody change the subject? Is the result credible? (For example, a claim that a child is kidnapped every 30 seconds in America is absurd, since that would be more than a million children per year.)

95.	The last one. Statisticians are mostly from mathematical or scientific backgrounds, which means we are on a quest for truth. Our clients may mangle, misuse, and abuse our conclusions, but we try very hard not to do that ourselves.

96.	Nobody knows. The statement is usually attributed to Mark Twain, although he himself credited it to Benjamin Disraeli.