STAT 301 Exam Review: Descriptive Stats, Normal Distribution, Inference and Regression

Terms in this set (30)

1. Mean vs median in a right-skewed distributionThe mean is pulled toward the long right tail, so mean > median. The median is resistant to outliers.
2. Sample standard deviation formulas = sqrt( sum of (x_i - xbar)^2 / (n - 1) ). It measures typical spread around the mean.
3. Why divide by n - 1 in the sample variance?Using n - 1 (degrees of freedom) makes the sample variance an unbiased estimator of the population variance.
4. Interquartile range and the 1.5 x IQR outlier ruleIQR = Q3 - Q1. A value is a suspected outlier if it is below Q1 - 1.5*IQR or above Q3 + 1.5*IQR.
5. Z-score formula and meaningz = (x - mu) / sigma. It tells how many standard deviations a value lies above or below the mean.
6. The 68-95-99.7 ruleFor a normal distribution, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
7. Mini-problem: scores are normal with mean 70 and SD 10. About what percent score above 90?90 is z = 2. About 95% lie within 2 SD, so about 2.5% lie above 90.
8. Standard error of the sample meanSE = sigma / sqrt(n), estimated by s / sqrt(n). Quadrupling n halves the standard error.
9. Central Limit TheoremFor a large enough sample size, the sampling distribution of the sample mean is approximately normal with mean mu and SD sigma/sqrt(n), whatever the population's shape.
10. Parameter vs statisticA parameter describes a population (mu, sigma, p) and is usually unknown. A statistic is computed from a sample (xbar, s, p-hat) and estimates the parameter.
11. General form of a confidence intervalEstimate plus or minus (critical value) x (standard error). For a mean: xbar plus or minus t* x s/sqrt(n).
12. 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% probability that this one interval contains it.
13. What makes a confidence interval narrower?A larger sample size, a lower confidence level, or less variability in the data.
14. When do you use t instead of z for a mean?When the population standard deviation is unknown and estimated by s. The t distribution has heavier tails and depends on degrees of freedom (n - 1 for one sample).
15. Null vs alternative hypothesisH0 is the no-effect or no-difference claim (e.g. mu = 50). Ha is what we seek evidence for (mu > 50, mu < 50 or mu not equal to 50).
16. Definition of a p-valueThe probability, assuming H0 is true, of getting a test statistic at least as extreme as the one observed. Small p-values are evidence against H0.
17. Decision rule with significance level alphaReject H0 if p-value <= alpha (often 0.05). Otherwise fail to reject H0; we never accept or prove H0.
18. Type I vs Type II errorType I: rejecting a true H0 (probability alpha). Type II: failing to reject a false H0 (probability beta). Power = 1 - beta.
19. One-sample t statistict = (xbar - mu0) / (s / sqrt(n)), with n - 1 degrees of freedom.
20. Paired t-test vs two-sample t-testPaired: two measurements on the same subjects (before and after); analyze the differences. Two-sample: two independent groups; compare the two means.
21. Assumptions of t proceduresRandom sample or randomized experiment, independent observations, and a roughly normal population or a large enough sample. They are sensitive to strong skew and outliers in small samples.
22. What does one-way ANOVA test?Whether three or more group means are all equal. F = (between-group variability) / (within-group variability); a large F suggests at least one mean differs.
23. Least squares regression liney-hat = b0 + b1*x, chosen to minimize the sum of squared residuals. Slope b1 = r*(sy/sx), and the line passes through (xbar, ybar).
24. Interpreting a slope of b1 = 2.5 for y = exam score and x = hours studiedEach additional hour of study is associated with a predicted increase of 2.5 points in exam score, on average.
25. ResidualResidual = observed y - predicted y-hat. A residual plot with no pattern supports a linear model.
26. What does r^2 mean?The fraction of variation in y explained by the linear regression on x. r = 0.6 gives r^2 = 0.36, so 36% of variation is explained.
27. Properties of the correlation rBetween -1 and 1, unitless, measures only linear association, and is strongly affected by outliers.
28. ExtrapolationUsing a regression line to predict far outside the range of observed x values. It is unreliable because the relationship may not continue.
29. Observational study vs experimentAn observational study measures without intervening and cannot prove causation. An experiment imposes treatments, ideally with random assignment and a control group, allowing causal conclusions.
30. Blocking in experimental designGrouping similar experimental units (e.g. by age) and randomizing treatments within each block to reduce variability from that factor.