Super AP Stats Prep

• On the AP Test, SAY WHAT THE VALUE THAT YOU FOUND IS before you explain the context. (ex: “r2 = 0.955. Context: 95.5% of the variation in ….”)

Semester 1 Review

Unit 1

• Use: Tables, Pie Charts, Dotplots, Bar Charts, Segmented/Side-by-Side Bar Charts

• Use: Histograms, Scatterplots, Dot Plots, Stem-and-Leaf plots, Boxplots

• Types of Variables: Discrete (Countable) vs. Continuous (not countable - weight)

• Shape: Modality (How many “Peaks” the data has - Unimodal, Bimodal, etc.) and Shape of Data (Symmetric, Skewed Left/Right, Uniform)

• Skewed Left: Tail of Data to the LEFT/ Mean pulled towards the left -> Median > Mean

• Skewed Right: Tail of Data to the RIGHT/ Mean pulled to the right -> Median < Mean

• In Symmetric or Unimodal distributions, Mean = Median

• Outliers: Use IQR Rule (1.5 * IQR) or use 2 S.D. Rule (For Normal Distributions, outliers exist outside 2 standard deviations of the mean -> μ± 2σ)

• Center: Use mean/median/Q1 or Q3 measures, approximate if necessary.

• Spread: Use Standard Deviation, Range, IQR, etc.

• What is an IQR: Q3-Q1, represents the spread/range of the middle half (50% of the data) of the data

• Is Standardized (Doesn’t take Units)

• Both Z-Scores and Percentiles can be used for ANY DISTRIBUTION

• Use NormalCDF to calculate Proportions. When using z-scores w/ Normalcdf, μ=0, σ=1.

• invNorm calculates the value OR z-score given the proportion (area).

Example Problems

1. Scientists working for a water district measure the water level in a lake each day. The daily water level in the lake varies due to weather conditions and other factors. The daily water level has a distribution that is approximately normal with mean water level of 84.07 feet. The probability that the daily water level in the lake is at least 100 feet is 0.064. Which of the following is closest to the probability that on a randomly selected day the water level in the lake will be at least 90 feet?

• To solve the probability, we’ll need to find the missing Standard Deviation. Find the Z-score of the proportion using invNorm(proportion, 0, 1) -> invNorm(0.064, 0, 1) = 1.522. Next, set the z-score equal to the z-score formula -> 1.522 = (100-84.07)/σ. You can now solve for S.D, and do the problem as normal.

Unit 2

• Describe scatterplots with Strength, Direction, and Form (ex: Strong/Moderate/Weak, Positive/Negative, Linear/Curved/No Pattern)

• Must Contain (x̄, ȳ)

• Equation: ŷ = a + bx, ŷ = predicted y, x = reg x ALWAYS DEFINE VARIABLES!!

• To Interpret Slope: “For every 1-unit increase in x, the y is predicted to change by b.”

• To Interpret y-Intercept: “When x is 0, the predicted value of y is a.”

• To Calculate Slope: b = r(Sy/Sx)

• Extrapolation: When you attempt to estimate values with the LSR line using x-values that are far outside the data range. DO NOT!!!

• Ranges from -1 to 1 (r=±1 -> Perfect Linear Relationship, r<0 -> Neg Association, r>0 -> Pos association - y MUST CHANGE WITH X!)

• Unaffected by changing units/axes (switching x and y), but is not resistant to outliers.

• Interpret w/ strength, direction, form (ex: strong, positive, linear)

• r2 = sum of residuals2, interpret w/ the sentence above.

• Calculate w/ sqrt(r2/(n-2)), interpret w/ the sentence above.

• Outliers: Doesn’t follow the pattern of data AND has a large residual compared to others (Much smaller/larger y value compared to others).

• High Leverage: Has a much smaller/larger x value compared to other data points.

• Only High Leverage Points are Influential. Points that are both HL And Outliers are the MOST influential.

Unit 3

• SRS (Label individuals with number, select “n” unique individuals, identify the individuals)

• Stratified Random Sampling: used to ensure all groups in population are represented proportionally. Divide population into strata (a group that share a characteristic relevant to the study), select SRS from each strata

• Cluster Sampling: Makes sampling a large/spread-out population easier. Divide the population into “clusters” (heterogeneous group of individuals that are already mixed), select SRS of the clusters, sample all from the selected clusters

• Systematic Random Sampling: Easiest of the “random” methods. Select every “kth individual.

• Multistage Sampling: Allows for control over multiple characteristics, giving you a representative sample of a large population. Simply use more than 1 method (ex: multiple stratified random samples)

• Using a representative sample allows you to make inferences about the population parameter.

• Undercoverage: Using a bad sampling method or a bad sampling frame (list from which the sample is chosen) can result in some groups of the population being left out of the sampling process; they are not represented.

• Sample Size: Using a small sample size can prevent representing the population well.

• Bad (Biased Sampling Methods)

• Response Bias: When individuals respond inaccurately to the sample. This can occur when the subject at hand is sensitive, and it can depend on how or who the question is asked to.

• Nonresponse: When individuals chosen from the sample don’t or can’t respond.

• Question-Wording: Leading questions (providing background information that shows bias towards an answer), open-ended questions (questions that are hard to answer, and therefore, hard to analyze), double negatives (can confuse individuals), word choice, overlapping options, etc.

• Observational Studies involve simply surveying a sample, and not imposing treatments. With observational studies, you can only show association (e.g. ___ tends to increase/decrease ____)

• Explanatory (Independent) Variable /Factor: The treatment(s)

• Level: Strength or Size of an explanatory variable

• Response (Dependent) Variable: The outcome we measure

• Experimental Units: The objects that receive the treatments - can be groups (ex: Container of Insects) or Subjects (human/animals)

• Treatment: Condition/Action applied to the experimental units

• Control: Efforts to minimize variability in the way subjects are obtained and treated

• Control Group: Group of subjects that receive no treatment/ a placebo treatment. Used as a comparison to the treatment groups, and help to eliminate lurking variables.

• Placebo: “Dummy” treatment that should have no impact.

• Placebo Effect: When subjects respond to a placebo due to a belief that they are receiving the true treatment.

• Blind: Subjects (and experimenters - double blind) don’t know what treatment they are receiving

• Randomization: Using random assignment of treatments to the subjects. This helps to reduce or eliminate confounding.

• Replication: Using many subjects and having many trials

• Statistical Significance: When an observed effect is so large that it could rarely happen by chance

• Results from lack of control. Use blocked designs or stratification.

• Results from bad experimental design

• Randomized Comparative Experiment: Randomly assign subjects to treatment groups, compare results together.

• Blocked Design: Stratifying subjects into groups (not random), assigning treatments to each group (random), comparing results of groups separately.

• Allows us to control/isolate lurking variables.

• Matched-Pairs Design: Assigning treatments to “paired” subjects - each subject in pair recieves different treatment, or each subject recieves both treatments in random order. Compare within pairs.

Unit 4

• If Mutually Exclusive: P(A ∩ B) = 0

• If Independent: P(B|A) = P(B)

• Two Events CANNOT be both disjoint and independent.

• Can either be discrete (set # of outcomes, use a histogram or table to show each outcome + probability of each, use 1-var-stats to get μ and σ (outcomes in L1, probabilities in L2)) or continuous (Use the normal distribution to obtain probabilities).

• Sample Space: Shows all possible outcomes of a chance variable, display it using a Probability Model

• Transform random variables:

• μa+bX = a+ b* μX | μX ± μY = μX±Y

• σa+bX = √(b2σX2) | σX ± σY = √(σX2 + σY2)

• Interpret μ: “The expected value of X in context is μ”

• Interpret σ: “On a randomly selected ___, X in context will vary from the expected count of μ by σ on average.”

• Binary (Only 2 Outcomes), Independent (Each trial independent - just specify), Trials (NO FIXED # of TRIALS), Success (Specify p = __)

• Let X = # of trials until success, use geometcdf/geometpdf

• Binary (Only 2 Outcomes), Independent (Each trial independent - just specify OR use 10% rule if random sample), Number of Trials (Specify # of trials, n = __), Success (Specify p = __)

• Let X = # of successes in the trials, use binomialcdf/binomialpdf

• When Using ≤ or ≥: subtract 1 from the k-value/ n-value

Example Problems

1. A mathematics competition uses the following scoring procedure to discourage students from guessing (choosing an answer randomly) on the multiple-choice questions. For each correct response, the score is 7. For each question left unanswered, the score is 2. For each incorrect response, the score is 0. If there are 5 choices for each question, what is the minimum number of choices that the student must eliminate before it is advantageous to guess among the rest?

• Find the Expected Num of points you can earn for every choice you eliminate. Remember, you score 2 points if you miss the question, so you have to keep finding the expected number of points until it’s greater than 2.

• Eliminate 1 Q: (¼ chance of getting it right * 7 points earned)+(¾ chance of getting it wrong * 0 points earned) = 7/4 < 2

• Eliminate 2 Q: (⅔ chance of getting it wrong * 7 points earned) + (⅓ chance of getting it wrong * 0 points earned) = 7/3 > 2, therefore eliminate 2 questions!!!

Semester 2 Review

Unit 5

• Can show how much sample results vary from the parameter on average (Standard Error)

• As n (size of each sample) increases on a sampling distribution: The Data gets closer together/ reduced spread of the distribution

• If a sample result is RARE (5% or less) based on a sample/sampling distribution: Then there IS convincing evidence against the claim (would be unusual to obtain proportions less than 5% - SIGNIFICANCE LVL)

• Means and Proportions are unbiased (μ = μx, p = μp̂), but the Standard Deviation of the sampling distribution is biased, depending on the sample size (σx = Sx/√n)

• State: P(p̂ >, <, ≥, ≤ value), define p̂ in context (ex: proportion of students in a random sample of n)

• Plan: Shape, Center, and Spread (Show Normality w/ np≥10, n(1-p)≥10, show that μp̂ = p, show 10% rule is valid, and then show σp̂ = √(p(1-p))/n)

• Do: Use normalcdf using μp̂ and σp̂ or μx and σx

• Conclude: There is a __ chance that (a random sample of n will have ___/ the difference between p1 and p2 will be ).

• State: P(x >, <, ≥, ≤ value), define x in context (ex: mean PSAT score of random sample of n students)

• Plan: Shape, Center, and Spread (Show Normality w/ CLT, show that μx1 = μ1, show 10% rule is valid and then show σx = √(Sx2/n)

• Do: Use normalcdf using μp̂ and σp̂ or μx and σx

• Conclude: There is a __ chance that (the mean of x1 from a random sample will be ___/ the difference in means of x1 and x2 will be ___).

Unit 6

• Increase Power by: increasing significance level, increasing sample size (reduces variability), and increasing the distance between your null hypothesis (Ho) and your specific parameter (this makes it easier to detect that the null doesn't equal the true value)

• To maximize power: pick the largest significance level you're willing to risk (remember, it increases the chance of a type 1 error), and the largest sample size you can afford

• Bigger Sample Size == smaller standard deviation of sampling distribution == minimum value to reject the null hypothesis decreases == power increases.

• To Calculate: (p̂ - p0)/ √ (p0(1-p0) / n) OR  (p̂1-p̂2 - (p1-p2) )/ √ ( (p̂C(1-p̂C) / n1 + (p̂C(1-p̂C) / n2 )

• TO CHECK CONDITIONS FOR A Z-TEST: USE np0, NOT np̂

• How to interpret: --% of all possible samples of the same size (n) will result in calculating an interval that contains the true parameter (in context)

• How to interpret: We are --% confident that the true parameter (in context) is contained in the interval point estimate +- margin of error

• To Calculate: Z* x √ (p̂(1-p̂) / n)

• Increase Margin of Error By: increasing confidence level (makes interval wider for a better success rate), increasing POPULATION standard deviation

• Decrease Margin of Error By: increasing sample size (reduces variability)

Example Problems

1. An experimenter conducted a two-tailed hypothesis test on a set of data and obtained a p-value of 0.44. If the experimenter had conducted a one-tailed test on the same set of data, which of the following is true about the possible p-value(s) that the experimenter could have obtained?

• To get the possible p-values from a one-sided test, given the p value from a 2 sided test, remember that the p value could differ from left/right sided tests. As such, you’ll want to (obviously) halve the p-value. This is your first possible value for the one-sided test’s p value. To obtain the second possible value, you have to do 1- p value of the one sided test. This gives you 1-0.22 -> 0.78 as the second possible value.

Unit 7

• As k increases, the normality increases and tail area decreases (smaller t* value)

• TLDR: non-normality of a population is fine in a t-distribution EXCEPT WHEN LARGE OUTLIERS AND HEAVY SKEW -> if CLT not met, use sample data graphs (N.P.P, Box Plot)

Unit 8

Unit 9

• Linear: Make Scatterplot Graph and Residual Graph, comment on them (AP: The true relationship between y and x is linear)

• Independent: 10% Rule OR Assume subjects/treatments independent

• Normal: Make N.P.P of Residuals (basically scatterplot with X = reg x, Y = residuals), check if linear (AP: The values of y are approximately normally distributed for each value of x)

• Equal Variance: Check residual graph for equal # of residuals above/below 0, and with random spread (AP: The standard deviation for y does not vary with x - As the value of the explanatory variable increases, the variability of the response variable stays the same (fails condition if scatterplot gets more varied as X increases) )

• Random: Normally given - just specify “random sample/assignment given”

• T-Tests: There is a pos/neg linear relationship between y and x.

• T-Intervals: We are __% confident that the slope of the population LSR line for y vs. x is contained in the interval _____

• For Collegeboard’s T-Intervals: We are __% confident that a ___ increase in x will result in a predicted increase in y of between _________ (Ex: “We are 98 percent confident that a 10-point increase in placement score will result in a predicted increase in starting salary of between $3,150 and $3,360.”)

• Two-Sided (B ≠ 0): “observing a test statistic as extreme as ___ or more extreme”

• One-Sided (B > 0, B<0): “Observing a test statistic of ___ or greater/smaller” (Depends on > or <)

• TLDR: P-value interpretation MUST BE CONSISTENT with Ha)

• Extrapolation: When you attempt to estimate a value outside of an LSR line’s range of data values (lol)

• r: Correlation Coefficient, interpret w/ strength, direction, shape (ex: strong, positive, linear)

• r2: Coefficient of Determination, interpret by saying: r2 of the variation in y_var is accounted for by the LSR line that uses x_var as the explanatory variable.