Statistics: Unlocking the Power of Data Lock 5 Exam 2 Review STAT 101 Dr. Kari Lock Morgan 11/13/12 Review of Chapters 5-9

Statistics: Unlocking the Power of Data Lock 5 Exam 2 In class Thursday 11/15 Cumulative, covering chapters 1-9 (but not 8.2 or 9.2… everything we have done so far in the course) Closed book, but allowed 2 double-sided pages of notes prepared by you You will need a calculator, and will need to know how to compute p-values for normal, t, chi-square, and F distributions using your calculator Practice exam and solutions to review problems available under documents on the course webpage

Statistics: Unlocking the Power of Data Lock 5 Tuesday Prof Morgan, 1 – 2:30 pm, Old Chem 216 Wednesday Prof Morgan, 2 – 3 pm, Old Chem 216 Prof Morgan, 4:30 – 5:30 pm, Old Chem 216 Heather, 8 – 9pm, Old Chem 211A Thursday Prof Morgan, 1 – 2:30 pm, Old Chem 216 Also, 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 Office Hours This Week

Statistics: Unlocking the Power of Data Lock 5 Was the sample randomly selected? Possible to generalize to the population Yes Should not generalize to the population No Was the explanatory variable randomly assigned? Possible to make conclusions about causality Yes Can not make conclusions about causality No Data Collection

Statistics: Unlocking the Power of Data Lock 5 Variable(s)VisualizationSummary Statistics Categoricalbar chart, pie chart frequency table, relative frequency table, proportion Quantitativedotplot, 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 boxplotsstatistics by group, difference in means Quantitative vs Quantitative scatterplotcorrelation, simple linear regression

Statistics: Unlocking the Power of Data Lock 5 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

Statistics: Unlocking the Power of Data Lock 5 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 Hypothesis Testing

Statistics: Unlocking the Power of Data Lock 5 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 p-value

Statistics: Unlocking the Power of Data Lock 5 Hypothesis Testing 1.State Hypotheses 2.Calculate a test statistic, based on your sample data 3.Create a distribution of this test statistic, as it would be observed if the null hypothesis were true 4.Use this distribution to measure how extreme your test statistic is

Statistics: Unlocking the Power of Data Lock 5 Distribution of the Sample Statistic 1.Sampling distribution: distribution of the statistic based on many samples from the population 2.Bootstrap Distribution: distribution of the statistic based on many samples with replacement from the original sample 3.Randomization Distribution: distribution of the statistic assuming the null hypothesis is true 4.Normal, t,  2, F: Theoretical distributions used to approximate the distribution of the statistic

Statistics: Unlocking the Power of Data Lock 5 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

Statistics: Unlocking the Power of Data Lock 5 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 Using Distributions

Statistics: Unlocking the Power of Data Lock 5 Confidence Intervals

Statistics: Unlocking the Power of Data Lock 5 Confidence Intervals Return to original scale with

Statistics: Unlocking the Power of Data Lock 5 Hypothesis Testing

Statistics: Unlocking the Power of Data Lock 5 General Formulas When performing inference for a single parameter (or difference in two parameters), the following formulas are used:

Statistics: Unlocking the Power of Data Lock 5 General Formulas For proportions (categorical variables), the normal distribution is used For inference involving any quantitative variable (means, correlation, slope), the t distribution is used

Statistics: Unlocking the Power of Data Lock 5 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)

Statistics: Unlocking the Power of Data Lock 5 ParameterDistributionConditionsStandard Error Proportion Normal All counts at least 10 np ≥ 10, n(1 – p) ≥ 10 Difference in Proportions Normal All counts at least 10 n 1 p 1 ≥ 10, n 1 (1 – p 1 ) ≥ 10, n 2 p 2 ≥ 10, n 2 (1 – p 2 ) ≥ 10 Meant, df = n – 1n ≥ 30 or data normal Difference in Means t, df = smaller of n 1 – 1, n 2 – 1 n 1 ≥ 30 or data normal, n 2 ≥ 30 or data normal Paired Diff. in Means t, df = n d – 1n d ≥ 30 or data normal Correlation t, df = n – 2n ≥ 30 pg 470

Statistics: Unlocking the Power of Data Lock 5 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 For testing for a difference in means across multiple groups, use ANOVA

Statistics: Unlocking the Power of Data Lock 5 Simple linear regression estimates the population model with the sample model: Simple Linear Regression

Statistics: Unlocking the Power of Data Lock 5 Simple Linear Regression Inference for the slope can be done using

Statistics: Unlocking the Power of Data Lock 5 Inference for the Slope

Statistics: Unlocking the Power of Data Lock 5 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) Intervals

Statistics: Unlocking the Power of Data Lock 5 Inference based on the simple linear model is only valid if the following conditions hold: 1)Linearity 2)Constant Variability of Residuals 3)Normality of Residuals Conditions for SLR

Statistics: Unlocking the Power of Data Lock 5 Inference Methods