Elementary Statistics - Collecting Data Study Guide

Topics covered: Chapter 1: collecting data - Structure of data - Sampling from a population - Experiments and observational studies ● Numerical – relating to numbers ● Categorical – relating to a group (gender) ● Cases – subjects or items that data are collected from. o Cases appear in rows, variables appear in columns ● Population – everyone ● Sample – part of population ● Explanatory variable – independent, predictors, X ● Response variable – dependent, outcome, Y ● Confounding variable – changes both variables at the same time ● Observational study: casual relationships cannot be established as there can be numerous unknown confounding variables. Watching something in its natural state and recording observations ● Experiments: clinical trials, manipulation of one variable Chapter 2: describing date - Categorical variables - One quantitative variable: shape and center - One quantitative variable: measure of spread - Boxplots and quantitative/categorical relationships - Two quantitative variables: scatterplot and correlation - Two quantitative variables: linear regression ● 1 quantitative (numerical) variable: o Dotplot, histogram, boxplot (best for identifying outliers in a quantitative variable) o Center: mean, median o Other locations: max, min, 1st and 3rd quartile o Spread: standard deviation, interquartile range, range, (distribution) ● 1 categorical variable: o Bar chart, pie chart o Summary statistics: frequency, relative frequency, proportion ● 2 categorical variables: o Segmented or side-by-side bar chart o Summary statistics: two-way table, row/column proportions, difference in proportions ● 1 categorical and 1 quantitative (numerical) variable: o Side-by-side boxplot, dotplot, histogram o Summary statistic: quantitative statistics within each category, difference in means ● 2 quantitative (numerical) variables o Scatterplot o Summary statistic: correlation, regression line One numerical: histogram, boxplot, dotplot ● Relative frequency = sample proportion ● Skewed to the left vs right: o Skewed to right: more of the dots are on the left side of graph o Skewed to left: more of the dots are on the right side of graph ● Calculate IQR: Q3-Q1 ● Residual is negative Chapter 3: confidence intervals - Sampling distributions - Understanding and interpreting confidence intervals - Constructing bootstrap confidence intervals - Bootstrap confidence intervals using percentiles ● Parameter: number that describes some aspect of a population ● Statistic: number that is computed from data in a sample Parameter (population) Statistic (sample) Mean (average) Mu μ x-bar Proportion p p-hat Standard deviation Sigma σ s Correlation Rho ρ r Slope β b ● In a p-hat dotplot, the center will be p ● In an x-bar dotplot, the center will be mu ● The standard deviation of the sample statistic is called the standard error (SE) of the statistic o SE is a measure of variability of the sample statistic. Higher SE implies higher variability ● For large samples (30 or more) the sampling distribution is a bell curve ● N(mean, standard deviation) – format for normal distribution, standardize statistic to have mean 0 and standard dev. 1 with z-scores o X~N(50,10) - mean 50, SE 10 ● Z = x – mean / sd ------ (statistic) - (null value) / (SE) ● Z = (x-mu)/SE ● Bootstrap sample: a sample created from a different sample being generated over and over ● 95% confidence interval formula: o Statistic +/- ME (margin of error) o Statistic +/- 2*SE (margin of error) ● A 95% confidence interval for mu: o x-bar +/- 2*SE ● A 95% confidence interval for p: o p-hat +/- 2*SE ● Bootstrap samples must: o Have the sample size as the original sample o Same measurement values as the ones used in the original sample ● Bootstrap distribution (histogram or dotplot): o Each dot represents the bootstrap sample statistic o Center of bootstrap distribution is the value of the original statistic o SE of the bootstrap distribution = SE of the sampling distribution ● Bounds of 95% CI o Upper bound: statistic (p-hat) + 2*SE o Lower bound: statistic (p-hat) – 2*SE ● Center of a sampling distribution = estimate of the parameter ● Center of a bootstrap distribution = statistic value of the original sample (which serves as a parameter estimate) ● Standard error of the statistic = the standard deviation of the sampling distribution = standard deviation of bootstrap distribution ● Bootstrap samples from original samples must contain the same “n” value, and be comprised of only the same numbers in the original sample Chapter 4: hypothesis tests - Introduction - Measuring evidence with p-values - Determining statistical significance - A closer look at testing - Making connections ● H0: null hypothesis, there is no effect/difference ● Ha: alternative hypothesis, claim for which we seek evidence o Ha: μ1 > μ2 (upper tail, right side) ▪ x-bar = 2 ▪ H0: μ = 0 ▪ Ha : μ > 0 o Ha: μ1 < μ2 (lower tail, left side) ▪ x-bar = -1 ▪ H0: μ = 0 ▪ Ha : μ < 0 o One sided alternative contains: > or < o A two-sided alternative contains: ≠ ▪ x-bar = 2 ▪ H0: μ = 0 ▪ Ha : μ ≠ 0 ▪ For a two-sided alternative, the p-value is twice the proportion of the smallest tail ● P-value: probability of obtaining a sample statistic as extreme as the observed sample statistic, when the null hypothesis is true o Small p-value: the stronger the evidence against the null hypothesis and in support of the alternative hypothesis ● If p-value < α, we reject H0 (alternative is true) ● If p-value ≥ α, we do not reject H0 ● α significance value usually = 0.05 unless otherwise specified ● Estimate p-value by a randomization distribution of sample statistics o If the original sample falls out in the tails, then a result that is that extreme is unlikely to occur if the null hypothesis is true. Evidence against null o Centered at the value of the parameter given in the null hypothesis ● ● If the p-value is smaller than the significance level, it is considered statistically significant Reject H0 Do not reject H0 H0 true Type 1 ERROR (rejecting a true null, false positive) - The probability of making a type 1 error is the significance level, α H0 false Type 2 ERROR (not rejecting a false null, false negative) ● Decrease α if type 1 error is very bad, increase α if type 2 error is very bad o Larger sample size makes it easier to reject the null ● The larger the sample size, the more compact the dots will be o (top is 250, middle 50, bottom 125) ● When approximating the population mean on a sampling distribution, it is the center of the graph ● When using percentiles of a bootstrap distribution and looking for a 95% CI, leave 2.5% on each side T-tables and Z-scores ● USE FORMULA SHEET and draw bell curves ● When asked to compute a 94% CI for the mean GPA of PA high school students using sample of 500 students ● For population mean (μ): 𝑥 ± 𝑧1− α σ/ 𝑛 > 𝑥 ± xth percentile * σ / 𝑛Given 2 ● Example problem when given: sample size (n), sample mean (weight), and population standard deviation (σ) and told to find 98% CI for actual weight o n = 49, large sample/top block of formula sheet o parameter of o 1. Draw out the bell curve, put 0.01 in left tail, .98 in middle portion, and 0.01 in right tail. You are looking for the portion up to the right tail, so on the table you are looking for the 99th percentile or as close to 0.99 as possible. ▪ Make the middle part the confidence level, then take the left tail too and find that percentile on the z table to use that value in the equation ● For a 94% CI, draw the bell curve, 0.03 | 0.94 | 0.03 , so on z-table, you are looking for 0.97 ● T-tables: for small sample sizes. You can look for the CI level on the bottom of the page. To find degrees of freedom, take n-1 and look on the table for that (n-1) value at the CI level. o Cum-prob = left proportion o 𝑥 ± 𝑡𝑛−1,1− α (𝑠/ 𝑛) > x-bar +/- t multiplier ( (𝑠/ 𝑛) 2 ● Finding minimum sample size: how many need to be sampled for average (weight) to be estimated within (0.5 lbs) with 98% confidence? 𝑧 o 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 (σ α 1− 2 ) parameter of interest: μ B is the margin of error, in this case = 0.5 N= 3* 2.33 (from z-table, get 99th percentile so 0.99 on table for the 98% confidence in the problem) / b= 0.5) ^2 ▪ n = (3*2.33/0.5)2 = (13.98)2 = 195.44 > 196 round up always Hypothesis testing using critical value method o μ0 = null value (center of x-axis on bell curve), α = significance level o reject H0 if p-value < α, reject H0 if the test statistic is more extreme than the critical value o Critical value is the z-score o The test statistic is: z = (x-bar – μ0) / (σ / √n) o Test statistic > critical value, reject H0 Ha: μ < #, critical region will be on the left of the graph. Ha: μ > #, critical region will be on the right of the graph. Ha: μ ≠ #, graph is two tailed therefore, +/- critical value is appropriate, divide α in two q-hat is 1- p-hat 𝐵 o o o ● ● ● ● ● Chapter 7: chi-square tests for categorical variables - Testing goodness-of-fit for a single categorical variable - Testing for an association between two categorical variables ● H0 specifies a proportion, pi, for each category o H0: p1=p2=p3=1/3 o Ha: at least one pi ≠ 1/3 ● Observed counts (frequency) are the amounts in each category actually observed, expected counts are the amounts in each category we should see if the null hypothesis is true o Expected = n * pi (in each category) ● Chi-square statistic: ● ● ● ● o Test statistic that compares the observed to the expected counts for all the cells o χ2 = Σ (observed-expected)2/expected to get test statistic Chi-square (χ2) distribution: o If each of expected counts are at least 5 And null hypothesis is true, then χ2 stat follows a χ2 – distribution, with degrees of freedom = number of categories – 1 o Always use the right tail to find a p-value Goodness of fit tests whether the distribution of a categorical variable is the same as some null hypothesized distribution. The null hypothesized proportions for each category do not have to be the same o Hypothesis that proportion of all majors are the same can be tested with goodness of fit Z-test statistic o Hypothesis that the proportions of science majors are more than the proportion of liberal arts majors can be tested with z test stat o Hypothesis that more than half the students are business majors is tested with z test Chi-square test of association for two categorical variables: o Hypothesis that there is no association between major and affinity to solar powered energy can be tested with test of association o Hypotheses, expected counts, chi-square statistic, chi-square distribution o Expected count = row total * column total / sample size o H0 means no association o Df= # rows -1 * # columns - 1 o Go to statkey to get p-value o For critical value go to stat key and use third option in theoretical distributions Chapter 8: ANOVA to compare several means - Analysis of variance - Pairwise comparisons and inference after ANOVA ● ANOVA compares the variability between groups to the variability within groups: total variability = variability between groups + variability within groups o If between group variability is larger than within group variability, it says whatever group characteristic varies plays a role? o k= number of groups o ni = number of units in group i o n = overall number of units o x-bari = mean for group i o x-bar = grand mean for combined data o H 0 : μ 1 = μ 2 = … = μk o Ha: At least one μ i ≠ μj (at least two means differ) ● If the groups are different, the variability between groups should be higher than the variability within groups ● Sums of squared deviations: o o SSTotal (total sum of squares) = SSG/B (sums of squares due to groups) + SSE (“error” sum of squares) Source Df Group s Error Total Sum of squares k-1 SSG n-k SSE Mean square (sum of squares divided by df) MSG = SSG / (k-1) MSE = SSE / (n-k) - Average variability F statistic p-value MSG / MSE Upper tail Fk-1, n-k n-1 SSTotal - Variability ● Compare two measures with f statistic o The f statistic is a ration of the average variability between groups to the average variability within groups o F = MSG / MSE o If there really is a difference between the groups, we would expect the f-statistic to be higher than we would observe by random chance o For the F statistic, the p-value is always the upper tail ● To find “within group (error) sum of squares” o ● ANOVA rules o The sums of squares and df values add up o MS = SS/df o Df for between groups = k-1, for total = n-1 Inference for Slope and Correlation ● Y-yhat is the residual y, the difference in the observed value and the predicted line value for that x value ● Population/true simple linear model o Y = β 0 + β1x + ε > intercept + slope + random error o β0 and β1 are unknown parameters o estimate with the… ● least squares line (prediction/regression equation) o yhat = b0 + b1x o yhat= intercept + slope estimate * x ● On minitab, o Constant is the intercept (intercept estimate) – first row ? (b0) o Coef is the x term (slope estimate) – second row ? (b1) o SE coef is standard error of each coef o H0: β1 = 0 vs. Ha: β1 ≠ 0 o t = b1 (sample slope) -0 (null value)/ SE o minitab gives a two sided p value, so if working with one-tail divide by 2 ● CI estimation for β1 o b1 +/- t * SE for CI o b1 and SE are in the 2nd row under Coef and SE columns and t-multiplier has (n-2) degrees of freedom o to get ^ t-multiplier o R-sq is the square of correlation, r value find sq root ● To test for no linear correlation between y and x: o H0: ρ = 0 vs Ha: ρ ≠ 0 o t = r – 0 / square rt of (1-r2) / (n-2) ● One tailed: right tail (positive correlation), o H0: ρ = 0 vs Ha: ρ > 0 o Rejection region is q3 and above (right side) ● One tailed: left tail (negative correlation) o H0: ρ = 0 vs Ha: ρ < 0 o Rejection region is q1 and below (left side) ● Two tailed: divide α When using α for a two tailed test, divide by 2 and put for each tail o Rejection region is both tails ● Correlation (ρ or r): o ρ (or r) close to +1 implies a strong upward linear trend o ρ (or r) close to -1 implies a strong downward linear trend o ρ (or r) near 0 implies NO linear trend. This means that there could be no relationship, or a non-linear relationship can exist Probability ● Probability of event A, P(A), is the long run frequency or proportion of times the event occurs o Probability is always between 0 and 1 and always refers to an event ● ● ● ● o P(A) = 1 means A will definitely happen o P(A) = 0 means A will definitely not happen Equally likely outcomes: if all possible outcomes are equally likely, the probability of an event is: o P(A) = number of outcomes in A / total number of outcomes P(A and B) is the probability both events will happen P(A or B) is the probability that either (or both) will happen Additive rule: P(A or B) o P(A or B) = P(A) + P(B) – P(A and B) o ● P(not A) = 1 – P(A) ● Conditional Probability o P(A if B) is the probability of A, if we know B has happened P(A|B) ▪ Probability of A given that B has happened o P(A if B) ≠ P(B if A) ▪ P(homosexual if male) = 0.04 ▪ P(male if homosexual) = 0.82 ● P(A if B) = P(A and B) / P(B) o P(A and B) = P(A if B) * P(B) o P(A and B) = P(A) * P(B if A) ▪ If A and B are independent, then P(B if A) = P(B) so: ▪ If A and B are independent: P(A and B) = P(A) * P(B) ● Events are disjoint or mutually exclusive if only one of the two events can happen (roll a die, getting a 5 or getting 4 – cannot happen at same time) ● If A and B are disjoint: o P(A or B) = P(A) + P(B) ● Events are independent if P(A if B) = P(A) o Whether B happens or not is not of any impact to the probability of A