1. STAT 101
Dr. Kari Lock Morgan
Exam 2 Review

2. Exam Details Wednesday, 4/2
Closed to everything except two double-sided pages of notes and a non-cell phone calculator
page of notes should be prepared by you – no sharing
Okay to use materials from class for your page of notes
Best ways to prepare:
#1: WORK LOTS OF PROBLEMS!
Make a good page of notes
Read sections you are still confused about
Come to office hours and clarify confusion
Cumulative, but emphasis is on material since Exam 1 (Chapters 5-9, we skipped 8.2 and 9.2)

3. Practice Problems Practice exam online (under resources)
Solutions to odd essential synthesis and review problems online (under resources)
Solutions to all odd problems in the book on reserve at Perkins

4. Office Hours and Help Monday 3 – 4pm: Prof Morgan, Old Chem 216
Monday 4–6pm: Stephanie Sun, Old Chem 211A
Tuesday 3–5pm (extra): Prof Morgan, Old Chem 216
Tuesday 5-7pm: Wenjing Shi, Old Chem 211A
Tuesday 7-9pm: Mao Hu, Old Chem 211A
REVIEW SESSION: 5–6 pm Tuesday, Social Sciences 126

5. Stat Education Center
Reminder: the Stat Education Center in Old Chem 211A is open Sunday – Thurs 4pm – 9pm with stat majors and stat PhD students available to answer questions

6. Two Options for p-values
We have learned two ways of calculating p-values:
The only difference is how to create a distribution of the statistic, assuming the null is true:
Simulation (Randomization Test):
Directly simulate what would happen, just by random chance, if the null were true
Formulas and Theoretical Distributions:
Use a formula to create a test statistic for which we know the theoretical distribution when the null is true, if sample sizes are large enough

7. Two Options for Intervals
We have learned two ways of calculating intervals:
Simulation (Bootstrap):
Assess the variability in the statistic by creating many bootstrap statistics
Formulas and Theoretical Distributions:
Use a formula to calculate the standard error of the statistic, and use the normal or t-distribution to find z* or t*, if sample sizes are large enough

8. Pros and Cons Simulation Methods PROS:
Methods tied directly to concepts, emphasizing conceptual understanding
Same procedure for every statistic
No formulas or theoretical distributions to learn and distinguish between
Minimal math needed
CONS:
Need entire dataset (if quantitative variables)
Need a computer
Newer approach

9. Pros and Cons Formulas and Theoretical Distributions PROS:
Only need summary statistics
Only need a calculator
More commonly used
CONS:
Plugging numbers into formulas does little for conceptual understanding
Many different formulas and distributions to learn and distinguish between
Harder to see the big picture when the details are different for each statistic
Doesn’t work for small sample sizes
Requires more math and background knowledge

10. Accuracy
The accuracy of simulation methods depends on the number of simulations (more simulations = more accurate)
The accuracy of formulas and theoretical distributions depends on the sample size (larger sample size = more accurate)
If the sample size is large and you have generated many simulations, the two methods should give essentially the same answer

11. Data Collection Was the explanatory variable randomly assigned?
Was the sample randomly selected?
Yes No
Yes No
Possible to generalize to the population
Should not generalize to the population
Possible to make conclusions about causality
Can not make conclusions about causality

12. Variable(s) Visualization Summary Statistics Categorical bar chart,
pie chart
frequency table,
relative frequency table, proportion
Quantitative
dotplot,
histogram,
boxplot
mean, median, max, min, standard deviation, z-score, range, IQR,
five number summary
Categorical vs Categorical
side-by-side bar chart, segmented bar chart
two-way table, difference in proportions
Quantitative vs Categorical
side-by-side boxplots
statistics by group, difference in means
Quantitative vs Quantitative
scatterplot
correlation,
simple linear regression

13. Confidence Interval
A confidence interval for a parameter is an interval computed from sample data by a method that will capture the parameter for a specified proportion of all samples
A 95% confidence interval will contain the true parameter for 95% of all samples

14. Hypothesis Testing
How unusual would it be to get results as extreme (or more extreme) than those observed, if the null hypothesis is true?
If it would be very unusual, then the null hypothesis is probably not true!
If it would not be very unusual, then there is not evidence against the null hypothesis

15. p-value
The p-value is the probability of getting a statistic as extreme (or more extreme) as that observed, just by random chance, if the null hypothesis is true
The p-value measures evidence against the null hypothesis

16. Hypothesis Testing State Hypotheses
Calculate a test statistic, based on your sample data
Create a distribution of this test statistic, as it would be observed if the null hypothesis were true
Use this distribution to measure how extreme your test statistic is

17. Distribution of the Sample Statistic
Sampling distribution: distribution of the statistic based on many samples from the population
Bootstrap Distribution: distribution of the statistic based on many samples with replacement from the original sample
Randomization Distribution: distribution of the statistic assuming the null hypothesis is true
Normal, t,2, F: Theoretical distributions used to approximate the distribution of the statistic

18. Sample Size Conditions
For large sample sizes, either simulation methods or theoretical methods work
If sample sizes are too small, only simulation methods can be used

19. Using Distributions
For confidence intervals, you find the desired percentage in the middle of the distribution, then find the corresponding value on the x-axis
For p-values, you find the value of the observed statistic on the x-axis, then find the area in the tail(s) of the distribution

20. Confidence Intervals

21. Confidence Intervals
Return to original scale with

22. Hypothesis Testing

23. General Formulas
When performing inference for a single parameter (or difference in two parameters), the following formulas are used:

24. General Formulas
For proportions (categorical variables) with only two categories, the normal distribution is used
For inference involving any quantitative variable (means, correlation, slope), if categorical variables only have two categories, the t distribution is used

25. Standard Error
The standard error is the standard deviation of the sample statistic
The formula for the standard error depends on the type of statistic (which depends on the type of variable(s) being analyzed)

26. Standard Error Formulas
Parameter
Distribution
Standard Error
Proportion
Normal
Difference in Proportions
Mean
t, df = n – 1
Difference in Means
t, df = min(n1, n2) – 1
Correlation
t, df = n – 2

27. Multiple Categories
These formulas do not work for categorical variables with more than two categories, because there are multiple parameters
For one or two categorical variables with multiple categories, use 2 tests (goodness of fit for one categorical variable, test for association for two)
For testing for a difference in means across multiple groups, use ANOVA

28. Chi-Square Test for Goodness of Fit
State null hypothesized proportions for each category, pi. Alternative is that at least one of the proportions is different than specified in the null.
Calculate the expected counts for each cell as npi . Make sure they are all greater than 5 to proceed.
Calculate the 2 statistic:
Compute the p-value as the area in the tail above the 2 statistic, for a 2 distribution with df = (# of categories – 1)
Interpret the p-value in context.

29. Chi-Square Test for Association
H0 : The two variables are not associated
Ha : The two variables are associated
Calculate the expected counts for each cell:
Make sure they are all greater than 5 to proceed.
Calculate the 2 statistic:
Compute the p-value as the area in the tail above the 2 statistic, for a 2 distribution with df = (r – 1)  (c – 1)
Interpret the p-value in context.

30. Analysis of Variance
Analysis of Variance (ANOVA) compares the variability between groups to the variability within groups
Total Variability
Variability
Between Groups
Variability
Within Groups

31. ANOVA Table Source Groups Error Total df k-1 n-k n-1 Sum of Squares
SSG
SSE
SST
Mean
Square
MSG = SSG/(k-1)
MSE = SSE/(n-k) F
Statistic
MSG
MSE
p-value
Use Fk-1,n-k

32. Simple Linear Regression
Simple linear regression estimates the population model
with the sample model:

33. Inference for the Slope
Confidence intervals and hypothesis tests for the slope can be done using the familiar formulas:
Population Parameter: 1, Sample Statistic: 𝛽 1
Use t-distribution with n – 2 degrees of freedom

34. Intervals
A confidence interval has a given chance of capturing the mean y value at a specified x value (the point on the line)
A prediction interval has a given chance of capturing the y value for a particular case at a specified x value (the actual point)

35. Conditions for SLR
Inference based on the simple linear model is only valid if the following conditions hold:
Linearity
Constant Variability of Residuals
Normality of Residuals

36. Inference Methods