STAT 311 Final Review: Study Design, Probability, Confidence Intervals and Hypothesis Tests

Terms in this set (31)

1. Observational study vs experimentAn observational study records data without assigning treatments, so it shows association only. An experiment randomly assigns treatments, allowing cause-and-effect conclusions.
2. What is a confounding variable?A variable related to both the explanatory variable and the response, making it impossible to separate their effects.
3. Simple random sample vs stratified sampleSRS: every group of n individuals is equally likely to be chosen. Stratified: divide the population into similar groups (strata) and take an SRS within each.
4. Three principles of experimental designControl (comparison groups and holding variables constant), randomization, and replication (enough units). Blocking can also reduce variability.
5. Mean vs median with a skewed distributionThe mean is pulled toward the tail; the median is resistant. In right-skewed data the mean is usually greater than the median.
6. Interquartile range and the 1.5 x IQR outlier ruleIQR = Q3 - Q1. Values below Q1 - 1.5*IQR or above Q3 + 1.5*IQR are flagged as potential outliers.
7. Standard deviation meaningRoughly the typical distance of observations from the mean. Sample formula: s = sqrt(sum((x - xbar)^2) / (n - 1)).
8. Properties of the correlation rr ranges from -1 to 1 and measures the strength and direction of a linear relationship. It has no units, is unaffected by changes of scale, and is sensitive to outliers.
9. Least-squares regression line formulasy-hat = b0 + b1*x, with slope b1 = r * (s_y / s_x) and intercept b0 = ybar - b1*xbar.
10. Interpret r^2 in regressionThe proportion of variation in y explained by the linear relationship with x. r^2 = 0.64 means 64% of the variation is explained.
11. What is a residual?Residual = observed y - predicted y-hat. A residual plot with no pattern supports a linear model.
12. Pitfall: extrapolationUsing a regression line to predict far outside the range of observed x values, which is unreliable because the relationship may not continue.
13. Addition rule for probabilityP(A or B) = P(A) + P(B) - P(A and B). For disjoint events, P(A and B) = 0.
14. Conditional probability and independenceP(A | B) = P(A and B) / P(B). A and B are independent if P(A | B) = P(A), equivalently P(A and B) = P(A)*P(B).
15. Conditions for a binomial settingFixed number of trials n, each trial success or failure, trials independent, and the same success probability p on each trial.
16. Binomial mean and standard deviationMean = n*p; SD = sqrt(n*p*(1 - p)). With n = 100 and p = 0.3, the mean is 30 and SD is about 4.58.
17. The 68-95-99.7 ruleIn a normal distribution about 68% of values lie within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
18. z-score formulaz = (x - mu) / sigma: how many standard deviations x is above or below the mean.
19. Central Limit TheoremFor large n, the sampling distribution of the sample mean is approximately normal with mean mu and standard deviation sigma / sqrt(n), whatever the population shape.
20. Standard error of a sample proportionSE = sqrt(p-hat * (1 - p-hat) / n). For a test, use the null value p0 in place of p-hat.
21. General form of a confidence intervalEstimate plus or minus (critical value x standard error). For a mean: xbar plus or minus t* * s / sqrt(n).
22. Correct interpretation of a 95% confidence intervalIf we repeated the sampling many times, about 95% of the intervals built this way would contain the true parameter. It is not a 95% chance this one interval contains it.
23. How do sample size and confidence level affect interval width?Larger n gives a narrower interval (SE shrinks with sqrt(n)). Higher confidence gives a wider interval.
24. What is a p-value?Assuming the null hypothesis is true, the probability of getting a result at least as extreme as the one observed. Small p-values are evidence against H0.
25. Type I vs Type II errorType I: rejecting a true null hypothesis (probability alpha). Type II: failing to reject a false null hypothesis (probability beta). Power = 1 - beta.
26. One-sample t statistict = (xbar - mu0) / (s / sqrt(n)), with n - 1 degrees of freedom. Used when the population SD is unknown.
27. Paired t-test vs two-sample t-testPaired: the same subjects measured twice or matched pairs, so analyze the differences. Two-sample: two independent groups, compare their means.
28. Chi-square test statisticChi-square = sum of (observed - expected)^2 / expected over all cells. Large values indicate the data differ from what H0 predicts.
29. Expected count in a chi-square test of independenceExpected = (row total x column total) / grand total. Each expected count should be at least about 5.
30. Mini-problem: n = 25, xbar = 52, s = 10, testing mu0 = 50. Find t.SE = 10 / sqrt(25) = 2, so t = (52 - 50) / 2 = 1.0 with 24 degrees of freedom.
31. Statistical significance vs practical importanceWith a very large sample, even a tiny, unimportant difference can give a small p-value. Always consider the size of the effect, for example through a confidence interval.