Ap Statistics Final Exam 2023 Graded A+

What is a statistic? Give several examples. - ANSWER-A number computed from data. Ex. A related example of a sample would be a group of 7th graders in the United States What is a parameter? Give several examples. - ANSWER-A number that describes a population. Ex. An example of a population is all 7th graders in the United States. What alternate meaning does the word parameter have in other mathematical disciplines? - ANSWER-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. What are the parameters of a uniform distribution? a normal distribution? a binomial distribution? a geometric distribution? a t distribution? a distribution? - ANSWER-Uniform: min and max [also need to know whether distrib. is discrete or continuous] Normal: and Binomial: n and p Geometric: p t: df : df Describe how to recognize uniform, normal, binomial, geometric, t, and distributions. - ANSWER-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 : continuous, always skew right Define range and describe how to find it. - ANSWER-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. Define IQR and describe how to find it. - ANSWER-IQR (interquartile range) = Q3 - Q1. Use STAT CALC 1 to get 5-number summary, then VARS 5 PTS 9 - VARS 5 PTS 7. Describe how to find outliers (a) in a column of data; (b) in a regression setting. - ANSWER-(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. In regression, what names are given to the x and y variables? - ANSWER-Explanatory, response. What does MSE mean? Is it a synonym for variance? - ANSWER-Mean squared error = pop. variance (mean squared deviation from the mean). Sample variance is different, since denom. is n - 1 instead of n. What does s.d. measure, and how is it computed? - ANSWER-Pop. s.d. ( ) 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, equals the square root of MSE (square root of pop. variance), and s equals the square root of sample variance. What special geometric meaning does s.d. have in a normal distribution? - ANSWER-In a normal distribution (required), the distribution curve is bell-shaped, satisfies the 68-95-99.7 rule, and has inflection points at . What is skewness? Give two examples of different ways to detect skewness. - ANSWER-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. 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. How does one recognize lack of normality? - ANSWER-Easiest way is look for a pattern that is not straight in NQP. You can use g.o.f. to test for departures from expected bin counts. What is the most common type of regression? - ANSWER-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.] Which is usually of greater interest, the LSRL slope or the LSRL y-intercept? Why? - ANSWER-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 name do we give to r? What does r mean? How do we compute r? - ANSWER-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). What name do we give to r2? What does r2 mean? - ANSWER-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). Is r affected by choice of units (e.g., mm, cm, inches, feet, light-years)? How about b0 and b1? - ANSWER-No; yes. Is r affected by choice of which variable is x and which is y? How about b0 and b1? - ANSWER-No; yes. How do we typically compute b0 and b1? What other ways are there? - ANSWER-STAT CALC 8, or with formulas 6 and 8 on first page of AP formula sheet. (Never use formula 5.) Describe a few interesting properties of the LSRL. - ANSWER-The point (xbar, ybar) is always on the LSRL, regardless of whether or not that point exists as a data point in the scatterplot. LSRL is not resistant to outliers. The reason is that none of r, sx, or sy are resistant, and the LSRL's slope is b1 = r (sy/sx). 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? - ANSWER-Resid. = (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. Give several examples of "good" and "bad" residual plots and what they should be telling us. - ANSWER-A good residual plot will show a medium-to strong correlation (make a straight line). A bad residual plot will not show a correlation (the dots will be scattered and will not make a straight line) Tell whether the following regression-related terms are synonyms: ____________ outlier and ____________ observation. If not, why not? - ANSWER-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. Interpret b0 and b1 for a layperson. - ANSWER-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 What do the letters r.v. mean? Give two examples, one that is ____________ and another that is ____________ . - ANSWER-Random variable (discrete or continuous). Ex. Discrete Random Variable= Response with Yes - No values, Maximum Number of adult that can fit into a car Continuous Random Variable= Time of day (12:31:24 p.m.), Temperature (60.31450 F, Weight (154.25 pounds), etc. If X is a(n) ____________ , then is calculated by ____________ and is known by two names: ____________ or ____________ . - ANSWER-r.v., , 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 ____________ . - ANSWER-r.v., variance, Var(X), , square root, Var(X), The mean of a ____________ equals the ____________ of the ____________ . Is this always true? What about for differences? - ANSWER-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? - ANSWER-sum, sum, variances; true only for independent r.v.'s; variance of difference (assuming indep. r.v.'s) equals sum of variances The s.d. of a ____________ multiple of X equals the ____________ times ____________ . Is this always true? - ANSWER-scalar (i.e., a constant), scalar, ; yes Describe how each of the following is affected by linear transformations: r, , , IQR, range. - ANSWER-r: no change 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. - ANSWER-Standardized (dimensionless) representation of a data point, in s.d.'s. Can always be computed, even if data set is non-normal. Use formula . Tells how many s.d.'s a data value is above or below the mean. In probability theory, a Venn diagram showing no overlap indicates that two ____________ are ____________ ____________ . Is this term a synonym for ____________ ? If not, explain the difference. - ANSWER-events, mutually exclusive; independence; no; independence of A and B means P(A|B) = P(A), which is not at all the same as Why do we care about probability? Is it merely of interest to casinos and misguided people who waste their money on state lotteries? - ANSWER-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." Explain what a ____________ distribution is. Give three examples, using the three test statistics that we care most about in AP Statistics. - ANSWER-Sampling distribution of or diff. of means: Follows z if is known (rare), otherwise t. Sampling distribution of : Really binomial, but almost normal if pop. is large, np 10, and nq 10. Sampling distribution of difference of proportions: Almost normal if pops. are large, n1p1 5, n1q1 5, n2p2 5, n2q2 5. 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? - ANSWER-s.e.; no idea; yes What does LOLN stand for? State it correctly and in one of the many ways in which people misconstrue it. - ANSWER-Law of large numbers. CORRECT: As , approaches p. (Sometimes stated as " as .") WRONG: If , then the proportion of successes will start to increase until we "catch up." (Or, if , 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 . 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. What does CLT stand for? State it correctly and in one of the many ways in which people misconstrue it. - ANSWER-Central limit theorem. CORRECT: Consider any population, not necessarily normal, having finite . As , the sampling distribution of approaches . 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. - ANSWER-P-value, test; principles of good experimental design In your own words, define each of the following and describe how it is determined or computed. test statistic - ANSWER-umerical summary of a data-set that reduces the data to one value that can be used to perform a hypothesis test. In general, a test statistic is selected or defined in such a way as to quantify, within observed data, behaviours that would distinguish the null from the alternative hypothesis where such an alternative is prescribed, or that would characterise the null hypothesis if there is no explicitly stated alernative hypothesis. z = (xBar - Δ) / (sx / sqrt(n)) P-value - ANSWER-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. alpha level - ANSWER-the maximum probability that you reject the null hypothesis when the null hypothesis is true. P(Type I error) - ANSWER-occurs when the null hypothesis (H0) is true, but is rejected. P(Type II error) - ANSWER-occurs when the null hypothesis is false, but erroneously fails to be rejected. power - ANSWER-robability that the test will reject the null hypothesis when the null hypothesis is false df - ANSWER-Each of a number of independently variable factors affecting the range of states in which a system may exist, in particular. A direction in which independent motion can occur. sampling error - ANSWER-incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. critical value - ANSWER-the value corresponding to a given significance level. m.o.e. - ANSWER-statistic expressing the amount of random sampling error in a survey's results. Explain the difference between confidence level and confidence interval. - ANSWER-The confidence level is the percentage at which 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 confidence interval are two numbers that you are 95% confident (confidence level) that something will happen. 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? - ANSWER-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. 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? - ANSWER-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. 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! Can H0 ever be proved? Why or why not? - ANSWER-We cannot prove H0. All we can do is judge whether the evidence against it is "sufficient to reject" or "insufficient to reject." Can Ha ever be proved? Why or why not? - ANSWER-We can sometimes gather overwhelming evidence that H0 can be rejected in favor of Ha. In the real world, even in