AP Statistics Course Review. Exploring Data Variables can be categorical or quantitative Discrete or...

AP Statistics

Course Review

Exploring Data• Variables can be categorical or quantitative• Discrete or continuous• For categorical data, we use bar charts• Numerical data can be displayed using a dotplot,

stemplot, box-and-whisker plot, histogram or cumulative frequency plot

• Remember histograms have no spaces (unless a category has none)

• Must include key with stemplot• Always label axes and make sure you read the axes

when interpreting a graph.

Commenting on a graph

• Shape: symmetric, skewed, unimodal, uniform• Center: Mean and median• Spread: Range, standard deviation, Iqr, gaps,

outliers (1.5x iqr) added to quartile

Effect of changing units

• Changing units will change measures of center and spread by the same ratio as the multiplier.

• Adding or subtracting the same constant will change measures of center in a similar manner but will not change measures of spread.

Trial Run 1

Scatterplots

• Bivariate, explanatory, response• Correlation coefficient (r) -1 to 1• R does not change when you switch x and y,

nor will it change when you multiply or add• Only measures strength of linear relationship• Affected by outliers• Lurking variables• Danger of extrapolation

• Coefficient of determination (r2)• Residuals (observed – predicted)• Influential points• Transformations

Trial run

Sampling

• Census, survey, experiment, observational study

• Parameter (population) statistic (sample)• Convenience, SRS, stratified, cluster,

systematic• Bias: undercoverage, nonresponse, response• Placebo, blind, randomization, replication,

confounding variable

• Experimental designs: completely randomized, blocks, matched pairs

Trial run

Probability

• Law of large numbers: long-term relative frequency gets closer to true freq. as # trials increases

• Disjoint (mutually exclusive): cannot occur simultaneously

• Mand and ort• Conditional probability:• Independence: knowing one has occurred

doesn’t change chance of the other

Probability distributions

• Matches all possible values of variable with probability of it happening

• All probabilities must be between 0 and 1• Total of probabilities must be 1• Mean: • Variance•

Binomial Random Variables

• Fixed number of trials, success or failure• P remains constant each trial• Each trial is independent• (nCr) pr (1-p)n-r

• Mean: np• Variance: np(1-p)

Geometric Random Variable

• Success or failure• P constant, each trial independent• How many times until ….• Probability k trials occur before …• p (1-p)k-1

Trial run

Combining Variables

• Mean (x+y) = mean (x) + mean (y)• Mean (x-y) = mean (x) – mean (y)• If independent: variance (x+y)= var(x)+var(y)

Normal distributions

• Z-score• Standardize endpoints, find area under curve

Trial run

Sampling distributions

• All possible random samples are taken and used to create a sampling distribution of the sample mean

• Standard dev. :

• Central Limit Theorem: as the size of an SRS increases, the shape of the sampling dist. tends toward normal

Hypothesis Testing

• Sample Proportion• Ho: • Ha:• Test Statistic• Pvalue• Assumptions: p is from a random sample• Sample size is large (np>10 and n(1-p)>10)• Sample no more than 10% of population

Sample Mean

• Ho:• Ha: • Test Statistic• P value• Assumptions: from a random sample• Sample size is large (>30) or population

distribution is approximately normal

Hypothesis Testing

• Difference in 2 sample proportions:• Ho: • Ha:• Test statistic:• P value• Assumptions: independently chosen random

samples or treatments were assigned at random to individuals

• Both sample sizes are large (np>10, n(1-p)>10 works for both of them

Hypothesis Testing

• Difference in two sample means• Ho:• Ha:• Test Statistic• P value• Assumptions: 2 sample are independently

selected random samples• Sample size large (>30) or population

distributions are approximately normal

Hypothesis Testing

• Paired t test comparing 2 population means• Ho: µd = hypothesized value

• Ha: µd < > ≠ hypothesized value• Test statistic:• Pvalue:• Assumptions: Samples are paired• Random samples from a pop. Of differences• Sample size is large (>30) or population distribution

of differences is about normal

Hypothesis Testing

• Chi-Square GOF• Ho:• Ha:• Test Statistic• P value• Assumptions: based on random sample• Sample size is large – every expected cell count

at least 5• Degrees of freedom?

Hypothesis Testing

• Chi-Square Test of Homogeneity or Independence (2 way table)

• Ho: There is no relationship between __and _• Ha: Ho not true• Test Statistic: • P value• Assumptions: independently chosen random samples or

random assignation to groups• All expected cell counts are at least 5• Degrees of freedom?

Hypothesis Testing (last one!!)

• Chi-square test for slope• Ho:• Ha:• Test statistic:• P value• Assumptions: dist. of e has mean value=0, std. dev.

of e does not depend on x, dist. of e is normal, random dev. of e are independent of each other

• Degrees of freedom: n-2

Confidence Intervals

• Statistic ± margin of error(also called bound)• Margin of error is combination of 2 numbers:

(Critical value ) (standard error)