Must Pass Quiz
Give the "approved wording" for a conclusion to a statistical test that does not show significance.
"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.
2-way (CSDELUXE or STAT TESTS C)
- independence: population greater than 10% of sample - Random: should say - counts: frequency greater than 5 and must be units of counts
1-sample t (STAT TESTS 2, 8)
- independence: population greater than 10% of sample - Random: should say - matched: or only one sample -differences are normal
Explain the following paradox: For a gambler to return from a casino as a winner is not rare, yet casinos are reliably profitable.
For the individual player who plays a few dozen hands of blackjack or pulls the arm on a slot machine a few dozen times, the sampling distribution of net outcomes is relatively short and wide, meaning that a good portion of the sampling distribution can spill into positive territory even though the mean is negative. This is why it is not rare for people to return home from Las Vegas as winners. (Fewer than half are this lucky, but since the lucky ones are usually the only ones who say anything, it is easy to get a false impression that winning is common.)
Define IQR and describe how to find it.
IQR (interquartile range) = Q3 - Q1. Use STAT CALC 1 to get 5-number summary, then VARS 5 PTS 9 - VARS 5 PTS 7.
What special geometric meaning does s.d. have in a normal distribution?
In a normal distribution (required), the distribution curve is bell-shaped, satisfies the 68-95-99.7 rule, and has inflection points at plus or minus sigma
What name do we give to r? What does r mean? How do we compute r?
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).
Give the "approved wording" for a conclusion to a statistical test that shows significance.
"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.
Give the "approved wording" for a conclusion to a confidence interval problem.
"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.
It has been said that 79.4% of all statistics are made up on the spot, that 5 out of every 3 Americans are weak at mathematics, that smoking is the leading cause of statistics, and that a statistician is someone who follows an unwarranted assumption to a foregone conclusion. Which of these flippant remarks is most unfair?
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.
Can H0 ever be proved? Why or why not?
We cannot prove Ho. All we can do is judge whether the evidence against it is "sufficient to reject" or "insufficient to reject."
Describe each step in the PHA(S)TPC process.
problem: state the problem hypothesis: wrote the null and alternative hypothesis based on the problem assumptions: check all required assumptions for the test you are about to do state: the formula for the test you will do test statistic: find the respected t or z score based on the information you are given p vale: assert the p value for the test statistic you found conclusion: state your conclusion based on your p value compared to your alpha level
Describe how each of the following is affected by linear transformations: r, , , IQR, range.
r: no change m : affected by both translation and dilation (fancy way of saying that m new= linear function of m old ) sigma: 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
The mean of a ____________ equals the ____________ of the ____________ . Is this always true? What about for differences?
sum, sum, means; yes; mean of difference equals difference of means
The variance of a ____________ equals the ____________ of the ____________ . Is this always true? What about for differences?
sum, sum, variances; true only for independent r.v.'s; variance of difference (assuming indep. r.v.'s) equals sum of variances
Describe how to use the result of #79 to get a formula for the s.e. of b1 that is much simpler than the one given on the AP formula sheet.
t = ( observed- expected)/ standard error= b1-0/ standard error= b1/ s of b1 in the LSRL t-test, s of b1= b1/t
Describe, in general terms, how the t statistic is calculated.
t= stat- parameter/ standard deviation of statistic
P-value
the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. you can find it by using the calculator or using a table with the z or t you found previously
g.o.f. (CSDELUXE)
- independence: population greater than 10% of sample - Random: should say - counts: frequency greater than 5 and must be units of counts
2-sample z (STAT TESTS 3, 9)
- independence: population greater than 10% of sample - Random: should say - large enough population: all p(n) greater than 10 and q(n) greater than 10
1-sample z (STAT TESTS 1, 7)
- independence: population greater than 10% of sample - Random: should say - large enough population: p(n) greater than 10 q(n) greater than 10
1-prop. z (STAT TESTS 5, A)
- independence: population greater than 10% of sample - Random: should say - large enough: p(n) greater than 10 q(n) greater than 10
2-prop. z (STAT TESTS 6, B)
- independence: population greater than 10% of sample - Random: should say -large enough population: p(n) greater than 10 q(n) greater than 10
2-sample t (STAT TESTS 4, 0)
- independence: population greater than 10% of sample - Random: should say -normal distributed: symmetric and unimodal
LSRL t-test (STAT TESTS E)
- linear: points in regression should be spread out - independent: no points following something - constant variably: standard deviations should be similar - normal distribution: symmetric and unimodal
Describe your thought process when deciding upon the type of statistical test (or interval) to use in various problems: 1-sample t, 2-prop. z, g.o.f., etc.
1 sample t: do you know sigma? is there one sample? 2 sample t: do you know sigma? is there two samples? 1 prop z: is it a proportion? is there only one set of data? you must know sigma 2 prop z: is it a proportion? are there two sets of data? you must know sigma chi squared gof: is it one sample? are you looking for the expected value of said sample chi squared for independence: are you testing a relationship? do you have two categories? do you have observed values? chi squared for homogenity: are you testing likeness? do you have two different sets of things? do you have observed values?
Describe a few interesting properties of the LSRL.
10.The point (xbar, ybar) is always on the LSRL, regardless of whether or not that point exists as a data point in the scatterplot. 9.S residuals = 0. (Recall, a residual is defined for each data point. Residual = y - yhat.) 8.In a LSRL residual plot, there must always be the same total of absolute lengths below the center line as above the center line. This is not true for other types of curve fitting (median-median, exponential, logistic, logarithmic, et
What is a statistic? Give several examples.
A number computed from data. ( sample, x bar, s, s^2)
What is a parameter? Give several examples.
A number that describes a population. ( parameter, m, sigma, sigma^2)
The AP formula sheet gives two versions of the s.e. for a 2-sample t situation (difference of ____________). Explain how to tell which one to use.
Always use the first one, never the second.
What alternate meaning does the word parameter have in other mathematical disciplines?
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.
Define the term bias and give several examples of types of bias.
Bias = any situation in which the expected value of a statistic does not equal the parameter being estimated. Selection bias refers to 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.
What does CLT stand for? State it correctly and in one of the many ways in which people misconstrue it.
Central limit theorem. CORRECT: Consider any population, not necessarily normal, having finite sigmas . As n to infinity , the sampling distribution of x bar approaches N( m, sigma/ square root of n) .
What name do we give to r2? What does r2 mean?
Coefficient of determination. Tells what portion of the variation in one variable can be explained by variation in the other. If r = .6 then 36% of the variation in y (or x) can be explained by variation in x (or y).
Describe how to transform an "interval format" C.I. into an "estimate m.o.e." format.
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. plus or minus M. Depending on the problem, "est." will be x bar , p hat , x bar 1 - x bar 2 , or p hat 1 - p hat 2
Explain the difference between confidence level and confidence interval.
Confidence level is a percentage, typically 95% or 99%. It is how confident you are that the true mean, proportion, or other statistic for which you are trying measure is between the two endpoints in the confidence interval. The numbers in the confidence interval are of the same units as the statistic you are trying to locate.
How does one recognize lack of normality?
Easiest way is look for a pattern that is not straight in NQP. you can use chi squared g.o.f. to test for departures from expected bin counts.
Describe how to find outliers (a) in a column of data; (b) in a regression setting.
Easiest way is to make modified boxplot, then TRACE to see the points (use arrow keys). Outliers are more than (1.5)IQR below Q1 or more than (1.5)IQR above Q3. (b)No rule of thumb—just judge visually. Outliers have "large" residuals.
In regression, what names are given to the x and y variables?
Explanatory, response.
True or false: If there are two columns of data in an experiment, then the situation calls for use of 2-sample procedures. Explain your answer.
False: If there are matched pairs, you really have only one sample (namely, a column of differences).
Is gambling rational?
In general, no. On rare occasions, a game may have a positive expected value for the player, but nobody should ever wager more than he can afford. For example, if the lotto jackpot is large enough, spending a few dollars on tickets may be mathematically rational, but spending hundreds of dollars is not. The vast majority of games of chance are a waste of time and money.
Interpret b0 and b1 for a layperson.
Interpret b0 and b1 for a layperson.
Why is it usually a very bad idea to use the word probability in any sentence involving confidence intervals? Is it possible to make a true statement that combines these terms?
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 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.
What is meant by the saying, "Statistical significance is not the same as practical significance"?
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.
What is skewness? Give two examples of different ways to detect skewness.
Lack of symmetry. Right skewness means the central hump leans out to the right, forcing mean > median, since mean is less resistant to extreme values. Left 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.
What does LOLN stand for? State it correctly and in one of the many ways in which people misconstrue it.
Law of large numbers. CORRECT: As n to infinity , p hat approaches p. (Sometimes stated as " x bar to m as n to infinity .")
What is the most common type of regression?
Linear least-squares. [It is not sufficient to say linear, because the LSRL is not the only type of linear regression. For example, there is the median-median line, which is useful in some situations and which is more resistant than the LSRL.]
Explain marginal and conditional probabilities. With what data (quantitative or categorical) are marginal and conditional probabilities usually computed?
Marginal probabilities = fractions involving row or column totals divided by grand total. Conditional probabilities = fractions involving individual cells divided by a row or column total. Both are usually concerned with categorical data in 2-way tables.
What does MSE mean? Is it a synonym for variance?
Mean squared error = population variance (mean squared deviation from the mean). Sample variance is different, since denominator is n - 1 instead of n.
Is the binomial parameter p the same as the P-value of a test? What symbol is commonly used as an equivalent for 1 - p? Would the AP graders understand this without further explanation?
No; q; yes.
Is r affected by choice of units (e.g., mm, cm, inches, feet, light-years)? How about b0 and b1?
No; yes
Is r affected by choice of which variable is x and which is y? How about b0 and b1?
No; yes
Who coined the saying, "There are three kinds of lies: lies, d_____d lies, and statistics"?
Nobody knows. The statement is usually attributed to Mark Twain, although he himself credited it to Benjamin Disraeli.
Does the m.o.e. of a statistic depend on the size of the population? Explain briefly, giving an example if possible.
Not really. For example, the m.o.e. (at a 95% confidence level) of a 1300-person poll will be about 3 percentage points, regardless of whether the poll is taken in California or in Wyoming. You do not need a larger sample to get the same accuracy in California, even though the population of California is about 34 million, more than 60 times larger than that of Wyoming.
Explain how odds work. In particular, given a probability P(A) expressed as a fraction, explain how to compute the odds in favor of the event as well as the odds against the event. Explain why "casino odds" never equal the mathematical odds.
Odds in favor = ratio of favorable to unfavorable outcomes. Odds against = ratio of unfavorable to favorable outcomes. For example, if p = P(A) = 4/13, then the odds in favor of event A are 4 to 9, and the odds against A are 9 to 4.
How does one prove causation?
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.
In experiments, probability arises at the end in the form of a ____________ computed from the ____________ statistic. Describe the three ______ __ ___ ________ _______ and briefly describe how you would implement them when designing an experiment of possible interest to you personally.
P-value, test; principles of good experimental design 1. Repetition- we want the experiment to be repeated across several units of research (animals, plants, ect.). So if I wanted to test the effectiveness of new rabbit food, I would feed multiple rabbits the new food and multiple rabbits the old food. 2. Random- we want the units to be chosen at random ( or blocked then chosen at random) for each treatment to decrease bias and variability. So we would number the rabbits and randomly select one number without replacement until all rabbits were in one of the treatments 3. Control- we want our experiment to be under control and attempt to control all elements that could inhibit our research or act as a confounding variable. So we would make sure the rabbits have only eaten the specific kind of food they were assigned. ie. only feed them said food for weeks and not give them anything else like grass
What does s.d. measure, and how is it computed?
Population s.d. (sigma) 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, sigma equals the square root of MSE (square root of population variance), and s equals the square root of sample variance.
What do the letters r.v. mean? Give two examples, one that is ____________ and another that is ____________ .
Random variable (discrete or continuous). Discrete: Size of shirt, Class period, Spots Continuous: Weight, Height, Time
Give several examples of "good" and "bad" residual plots and what they should be telling us.
Random-looking residual plots are desirable. Common LSRL residual plot problems: Flange: The s.d. of the residuals changes with x. In other words, the amount of vertical "scattering" changes noticeably for different values of x.
Define range and describe how to find it.
Range is a single number for the spread of values in a column of data: range = max - min. Not 48 to 78.
Tell whether the following regression-related terms are synonyms: ____________ outlier and ____________ observation. If not, why not?
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.
What is a residual? How does one make a residual plot? If a residual plot for a LSRL model has residuals on the y axis, what variable goes on the x axis?
Resid. = y - y hat (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.)
Which assumption is more important, normality (if applicable) or the assumption that data come from an SRS? Why?
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.
How do we typically compute b0 and b1? What other ways are there?
STAT CALC 8, or with formulas 6 and 8 on first page of AP formula sheet. (Never use formula 5.)
Explain what a ____________ distribution is. Give three examples, using the three test statistics that we care most about in AP Statistics.
Sampling distribution of x bar or diff. of means: Follows z if sigma is known (rare), otherwise t.
What do the letters SRS stand for, and what is an SRS?
Simple random sample; a sample in which every possible subset is equally likely to be selected
Which is usually of greater interest, the LSRL slope or the LSRL y-intercept? Why?
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.
What is the purpose of a z score? Under what circumstances may one compute a z score? Describe how to compute it and what it means.
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)/ sigma . Tells how many s.d.'s a data value is above or below the mean.
Why do we care about probability? Is it merely of interest to casinos and misguided people who waste their money on state lotteries?
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."
The AP formula sheet gives two versions of the s.e. for a 2-prop. z situation (difference of ____________). Explain how to tell which one to use.
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.
Which is usually preferred: a one-tailed test or a two-tailed test? When should the decision be made regarding the type of test? What is the relevant question to consider in determining whether to use a one-tailed or two-tailed test?
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.)
Describe how to recognize uniform, normal, binomial, geometric, t, and distributions.
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 chi square: continuous, always skew right
What are the parameters of a uniform distribution? a normal distribution? a binomial distribution? a geometric distribution? a t distribution? a distribution?
Uniform: min and max [also need to know whether distribution is discrete or continuous] Normal: m and sigma Binomial: n and p Geometric: p t: df chi square: df
Give several examples of ways in which people lie with statistics.
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.
Can Ha ever be proved? Why or why not?
We can sometimes gather overwhelming evidence that H0 can be rejected in favor of Ha. 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.)
Give several examples of questions you should always ask when hearing or reading a statistic for the first time.
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.)
Is poker a game of chance?
Yes, but it is much more accurate to call poker a game of psychology and applied mathematics. Like pure games of chance, poker is an effective way to waste a great deal of time and money.
There are four types of employees at XYZ Corp., whom we will call pitchers, catchers, infielders, and outfielders for lack of a more creative idea. All categories of employees have recently had large cuts in their mean salaries, and yet the overall mean salary per employee has risen. Is such a thing possible? Explain.
Yes; perhaps many new employees have been hired.
There are four types of employees at XYZ Corp., whom we will call pitchers, catchers, infielders, and outfielders for lack of a more creative idea. All categories of employees have recently had large cuts in their mean salaries, and yet total payroll costs have risen. Is such a thing possible? Explain.
Yes; perhaps many new employees have been hired.
critical value
a cutoff value that determines the boundary between a test statistic being rejected against the null hypothesis and those that lead to a decision not to reject the null hypothesis. you can find it by inverse norm for z: ie 95% confidence= .975 or inverse t: ie for 95% confidence and population of 9= .975 and 8 df
test statistic
a test statistic is a sample used to determine whether a hypothesis of a population estimate will be rejected or failed to be rejected. Its usually given
P(Type I error)
a type I error occurs when you reject a true null hypothesis. its determined if you find the pvalue is less than alpha level and told it isnt.
P(Type II error)
a type II error occurs when you fail to reject a false null hypothesis. its determined by finding a pvalue greater than the alpha level and told it isnt.
Explain what is meant by double blinding, and why it is so important in clinical trials.
both the subjects and the giver of the treatment dont know which person is getting what treatment. this cuts down the bias of the experimenter and the people
Data from a small sample, from a person's own experience, or from a ____________ sample should usually be dismissed on the grounds that they are ____________ . However, data from large samples (for example, responses to on-line surveys or magazine subscriber surveys) are also often worthless. Why?
convenience, anecdotal; voluntary response bias
There is a popular saying involving correlation (more generally, association) and causation. What is the saying, and what does it mean?
correlation doesnt equal causation. just because its associated doesnt mean it causes something 100%
In probability theory, a Venn diagram showing no overlap indicates that two ____________ are ____________ ____________ . Is this term a synonym for ____________ ? If not, explain the difference.
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 and B)= 0
The purpose of ____________ statistics is to ___ ____________ ___ ____________ ____________ . (This is a much more difficult and sophisticated skill than descriptive statistics, in which we assume that any reasonably intelligent person should be able to read a table or a graph, compute s.d., add a LSRL trend line, etc. Be sure you explain this to people if they pooh-pooh your having spent a year studying statistics. There is much more to the subject than learning about means, modes, and medians!)
inferential, use statistics to estimate parameters
What is meant by statistical significance?
it is unlikely to have occurred by chance therefore there must be an association
df
its how many values that are free to vary. You can find it by k-1 or (r-1)(c-1) for two items
sampling erro
its the expected amount of variability from a sample. you can find it by sigma/ square root of the sample size
If X is a(n) ____________ , then is calculated by ____________ and is known by two names: ____________ or ____________ .
random variable., the sum of x1(p1) , mean, expected value
If X is a(n) ____________ , then ____________ is calculated as probability-weighted MSE and is indicated by either of two possible notations: ____________ or ____________. The ____________ ____________ of ____________ equals s.d., denoted ____________ .
random variable., variance, Var(X), (sigma of x)^2 , square root, Var(X), sigma of x
What abbreviation is sometimes used for the s.d. of a statistic? Why does the AP generally avoid this term? Would they understand us if we used it?
s.e.; no idea; yes
The s.d. of a ____________ multiple of X equals the ____________ times ____________ . Is this always true?
scalar (i.e., a constant), scalar, sigma of x ; yes
level
the alpha level is the ruler that measures your z or t value based upon the pvalue. Usually you are given the alpha level, can calculate it by taking confidence interval - 100, if not you can assume .05 (95% confidence)
m.o.e.
the amount of random sampling error from a survey. ME= critical value( s/ square root of population) or ( square root of p times q/ population)
power
the probability that the test will reject the null hypothesis when the null hypothesis is false (i.e. the probability of not committing a Type II error, hence the probability of supporting the alternative hypothesis when the alternative hypothesis is true). The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis. As the power increases, the chances of a Type II error occurring decrease. Therefore power is equal to 1 − β. Power increases with alpha level and population.
It can be proved, after a page or so of messy algebra, that s2 is an unbiased estimator of . (Curiously, though, s is not an unbiased estimator of .) Describe the two other unbiased estimators we learned about during the year.
x bar is an unbiased estimator of m ; i.e. E( x bar)= m p hat is an unbiased estimator of p; i.e. E( p hat)= p