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AP Stat Study Guide Detailed Questions And
Expert Answers
Difference between Population and Sample - ANS Both can be measured by a census. Population is a group of the individuals you want information about. Sample is a subset of individuals in the population that you actually collect data from. What do Samples allow you to do? - ANS Unlike Populations, the allow you to draw conclusions about whole populations based on the sample. Bias - ANS Anything that causes a sample to be not representative of the population of interest.
- Need to articulate what bias is, why it should be considered bias, and how it distorts results from what they might otherwise be. Difference between Sampling Error and Sampling Bias - ANS Sampling Error is ____ , however Sampling Bias is using a method that favors some outcomes over another. How does a small sampling size affect the validity of the sample? - ANS A smaller sampling size negatively affects the validity of the sample because the sampling error increases.
Types of Sampling Bias - ANS - Under-coverage Bais
- Non-response Bias
- Voluntary Response Bias Under-coverage Bias - ANS Occurs when some members of the population are inadequately represented in the sample.
- Often a problem in convenience samples Non-response Bias - ANS Occurs when there is failure to obtain a measurement on one or more study variables for one or more elements K selected for the survey. Basically when someone does not answer the question.
- A form of non-observation present in most surveys Voluntary Response Bias - ANS Occurs when sample members are self-selected volunteers, as in voluntary samples.
- In survey sampling - voluntary Types of Response Bias - ANS - Loaded Questions
- False Answers (Response Bias) Sampling Bias - ANS A bias introduced by who was included in the sample.
Multistage Sample - ANS Taking the samples in stages using smaller sampling units at each stage. Stratified Random Sample - ANS A method of sampling from a population which can be partitioned into sub-populations
- Will reduce the variability of possible sample results Convenience Sample - ANS A type of non-probability sampling method where a sample is taken from a group of people to contact or reach. How to design a random sampling procedure - ANS Describe a method
- Assign each (unit, subject, etc.) a DIFFERENT number between __ and __.
- Describe how you will implement the sampling method you want to use.
- Randomly select __ numbers, IGNORING REPEATS, and include the (unit, subject, etc.) that corresponds with those numbers in your sample. Difference between an Experiment and an Observational Study - ANS An observational study is a study where researchers (No control over the variables) simply collect data based on what is seen and heard and infer based on the data collected. An experiment is, however the applying treatments to a group and recording the effects. Treatment - ANS A control to the subjects that is imposed
- Controls their environment
Confounding - ANS A variable that influences both the dependent variable and independent variable causing a spurious association. Experimental Units - ANS It is the physical entity which can be assigned at random to a treatment.
- Subjects when it is an individual unit
- Commonly an individual unit
- Unit of statistical analysis 4 Principles of Good Experimental Design - ANS - Randomization
- Replication
- Control
- Stratification Control Group - ANS In an experiment, the group that is not exposed to the treatment; contrasts with the experimental group and serves as a comparison for evaluating the effect of the treatment.
- Not mandatory
- Kept the same throughout the experiment Placebo Effect - ANS Something that is expressed as one thing but it is really not that.
- A fake or placeholder
RBD ("Blocking") Advantages - ANS Advantages: Flexibility; can have any number of treatments and blocks, more accurate results; allows for calculation of unbiased error for specific treatments. RBD ("Blocking") Disadvantages - ANS Disadvantages: Not suitable for large numbers when complete block contains considerable variability; increased error. Matched Pairs Design - ANS Used when experiment has only two treatment conditions and subjects can be grouped into pairs. Matched Pairs Design Advantages - ANS Advantages: Fewer participant variables, no order effects, lower risk of demona characteristics, come tests/materials can be matched in every level. Matched Pairs Disadvantages - ANS Disadvantages: Participants cannot be matched in every level. Generalizability - ANS The extent to which the results of a sample (or experimental group can be applied to a certain population).
- You can generalize to the population from which the sample or experimental group was TAKEN
- Bias can hurt (or even eliminate generalizability); you need RANDOMNESS to avoid this 5 Things to Discuss while Analyzing the Distribution of Data - ANS 1. Center
- Spread
- Gaps
- Shape
- Outliers Center - ANS How to find it: Mean = Add up all data values/Total number of values Median = Middle data pt. 2 Types of measure:
- Mean (Sensitive to outlier effects)
- Median (Resistant to outlier effects) Best to usually rely on the mean, unless the data is skewed, then use median. Shape - ANS 5 Different ones:
- Normal (Mean = Median)
- Skewed Left (Mean > Median)
- Skewed Right (Median > Mean)
- Uniform (Mean = Median)
Boxplot - ANS Shows Min, Q1, Med, Q3, & Max Cannot show shape Outliers marked with an * Stemplot - ANS Have a Key to show what certain numbers mean Do not skip stems Read Stem first then leaf Dotplot - ANS Amount accounted for by a dot above x axis labeled Easy Histogram - ANS A graph of vertical bars representing the frequency distribution of a set of data. X axis shows INTERVALS, y axis shows FREQUENCY (the number of data points that belong in that interval) To find median: Find total amount of data points, use n+1 / 2 to find the position of the median, then see what interval contains that position. ^ Interval is your answer ^ Normal Distribution - ANS There is a symmetrical spread of frequency data that forms a bell shaped pattern. Theoretical, because in reality we do not consider data to be normal.
- Like a histogram where the center = mean and the intervals are one Standard Deviation each
Empirical Rule - ANS The rules gives the approximate % of observations within
- 1 standard deviation (68%) of data
- 2 standard deviations (95%) of data
- 3 standard deviations (99.7%) of data... of the mean when the histogram is well approx. by a normal curve Z-score - ANS Interpret: A measure of how many standard deviations you are away from the norm (average or mean). Calculate: Z = (your score - mean) / standard deviation W/ Calculator: Given Percentage % or probability use InvNorm Looking for Percentage % or probability use NormalCDF
- They can help us compare two unlike measurements Probability - ANS It is the possibility of an outcome occurring at an instance. Calculate Probability - ANS P(any event) = number of favorable outcomes / number of possible outcomes Law of Large Numbers - ANS The sample mean will approach the proportion mean. You get the expected value or numbers that you expect.
Dependent P(A & B) = P(A) * P(A | B) conditional probability Using nCr function - ANS Answering the question: "From a set of different item, how many ways can you select and order(arrange) these items?" "How many ways?" (Independent -Order does not matter) Conditional Probability - ANS You are given additional information to figure out the likelihood of an event occurring under that circumstance. P(A | B) = P(A n B) / P(B) Discrete Random Variable - ANS A random variable that takes on one of a list of possible values; typically counts. Finite Continuous Random Variable - ANS A random variable that may assume any numerical value in an interval or collection of intervals. Probability of getting exactly one given outcome = 1. Calculate Expected Value - ANS E(x) = x value (Probability) + .... Multiple times repeated with each value. Calculate SD (spread) by hand - ANS Variance is Standard Deviation Squared Var(x) = SD^ Var= x value (n - SD)^2 + ....
n = when it occurs Transforming and Combining Random Variables - ANS + or - Constant Center (Mean): Adds or subtracts Spread (SD): No change
- or / Constant Center (Mean): Multiplies or Divides Spread (SD): Multiplies or Divides Combining (Adding or subtracting two random variables to each other) Center (Mean): Changes Spread (SD): No change Expected Value - ANS The mean or average of a discrete random variable. (weighed by probability) Binomial Distribution - ANS ONLY APPLIES WITH CONDITIONS MET! B: Probabilities of sucesses are constant p = 3 I: Independent Observations -Knowlege of the outcomes of previous trials does not affect the PSuccess of another N: Number of observations is fixed -n=? S: Success or failure -All observations divided into these two outcomes
Shape of a Geometric Distribution - ANS Unimodal and Skewed always As you continue, the probability of having a success gets smaller. Explanatory Variable - ANS Helps to predict or explain changes in a response variable (x) Response Variable - ANS Measures the outcome of a study (y) Characteristics of Bivariate Data - ANS Shape: Possibilities -Unimodal, Uniform, Bimodal/nomial, Skewed(Right or Left) R Values -Assumes that shape is uniform Strength: Possibilities -Strong, Moderate, Weak, None & Linear R Value -Closer to 1 = Stronger, Closer to 0 = Weak(-1) Direction: Possibilities -Positive or Negative R Value -Positive or Negative Slope Outliers: (especially if they substantially alter the equation of the regression line, or line of best fit. Context: Always -what two variables are we examining? x and y correlation - ANS x does not cause y, doesn't imply causeation
Line of Best fit Equsn. - ANS Y(hat) = mx + b Definition of Y(hat) - ANS The predicted value of y for a given value of x. Interpretation of Slope (m) - ANS For every increase in x the model predicts on average that the y will increase by m Interpretation of Y-intercept - ANS When the value of x is at zero y is predicted to start at b(yint) r^2 Value ("coefficient of determination") - ANS This accounts for the amount of data explained by the SRS, LSRL Extrapolation - ANS The use of a regression line for prediction that is outside of the interval of values the explanatory x variable used to obtain the line. Residual - ANS The difference of an actual value to the predicted value in the LSRL Sign is an r Calculating a Residual - ANS Predicted Y value (Y hat) calculated - actual value = r Residual Plot - ANS Gives you information of the relationship of the actual data to the predicted data, how far you are off (difference) and if there is pattern/no pattern
Sampling Distribution P Hat - ANS Describes the distribution of values taken by the sample proportion P Hat in all possible samples of the same size of population. Central Limit Theorem - ANS CLT The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution. Sample Distribution - ANS A graph of data taken from one sample. Sampling Distribution - ANS A graph of statistics taken from multiple samples. CLT Conditions - ANS Only go through the process if the problem does not say "assume all conditions are met" Sample size: The Sampling distribution is normal
