# Inferential Statistics

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1 Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods

2 Inferential Statistics Descriptive statistics are useful on their own, but are also important as basis for making inferences from a sample of observations to characteristics of the population from which the sample came Mean of a sample can be used to suggest the likely value of the mean of the population Standard deviation of a sample can be used to suggest the likely value of the standard deviation of the population

3 Inferential Statistics Statistical inference Enables use of one or more samples of observations to infer values of a population Test hypotheses to determine if observed differences between groups or variables are real or occur simply by chance Produces new information by making predictions and generalizations based on samples Uses data to make estimates or inferences about population based upon sample Parameter estimating - estimating a parameter based on sample

4 Inferential Statistics Basic assumption: by carefully following certain scientific procedures, one can make inferences about a large group of elements by studying a relatively small number selected from the larger group Based on random sampling, probability theory, normal distribution

5 Inferential Statistics Population and sample Population or universe group of people, things or events having at least one trait in common Must be clearly defined, specifically delimited, carefully chosen Parameter is any measure obtained by measuring entire population Sample subset or subgroup of the population Statistic is any measure obtained by measuring sample

6 Sampling Population: Set of entities in which researcher is interested, with purpose of being able to describe certain characteristics or make other predictive statements Why not study whole population? Logistically impossible Infinite populations Future populations Destructive Too expensive Too time-consuming

7 Sampling Sample: Subset of a population that is examined and from which inferences are drawn about characteristics of population Benefits of sampling: Reduced cost Results available sooner Broader scope yields more information Greater accuracy More attention possible to each observation For example, compare results from expending effort to collect 1000 observations in either of the following ways: 1 variable observed on each of 1000 elements in the population 10 variables observed on each of 100 elements in a sample

8 Sampling Much of inferential statistics is based on the sample: Describing how accurately the sample represents the population Making inferences about the population Making predictions about the population Techniques of obtaining a valid sample are important

9 Sampling Determining sample size Sample size is directly related to power and effect size Power: the ability to detect real differences Number of participants in a sample is directly related to the standard deviation of the sample data set More participants narrower the distribution more likely any differences that exist will be detected Effect size: strength of the association between variables and/or the strength (size) of the difference found (small =.20, medium =.50, and large =.80) Greater the effect size greater the power

10 Sampling How large a sample is necessary for adequate power? Serious statistical texts have formulas to calculate actual and estimated effect sizes Rules of thumb vary by type of test to be used T test, analysis of variance: given a medium-to-large effect size, 30 participants per cell should lead to about 80% power Correlation or regression: at least 50 participants, number increasing with larger numbers of independent variables Chi-Square: no expected frequency in a cell should drop below 5, and overall sample should be at least 20

11 Normal Distribution Family of frequency distributions that have same general shape Symmetrical More scores in the middle than in the tails Bell-shaped curve Height and spread of a normal distribution can be specified mathematically in terms of: Mean Standard deviation

12 Normal Distribution If a set of data is normally distributed, it exhibits particular probability properties: Mean, median, and mode are very similar Curve is symmetrical on either side of mean value Probability of any value being above, or below, the mean value is 0.5 Any individual value is more likely to be closer to the central tendency than the extremes of the curve There is a constant relationship between the standard deviation and probability

13 Normal Distribution

14 Normal Distribution

15 Normal Distribution % 34% 13.5% 13.5% SD 1 SD Mean = 6 1 SD 2 SD SD = 2

16 Z-Score Standard normal distribution has a mean of 0 and a standard deviation of 1 Any normally distributed data set, which may very likely have different mean and standard deviation values, can be transformed to the standard normal distribution through the use of z scores Z score: where: x is a value from the original normal distribution x is the mean of the original normal distribution s is the standard deviation of the original normal distribution Z score thus becomes the number of standard deviations that a given value lies above or below the mean Relative value of score compared to the sample from which it comes

17 Z-Score Why convert data into Z scores? Enables comparing data between different distributions or two different variables Determines place or relative standing of a given score Example: Mary got 78 on Math, 115 on Science, and 57 on English portions of a national achievement test - how did she perform relative to others? Math Science English x x x = 75 = 103 = 52 S = 6 S = 14 S = 4 X = 78 X = 115 X = 57 z = 0.5 z = 0.86 z = 1.25

18 Z-Score Using standard normal distribution probability table enables stating z score as a relative percentage and to predict probability of any given score If z is positive, add.5 to table score to determine percentage below z score Math Science English x x x = 75 = 103 = 52 S = 6 S = 14 S = 4 X = 78 X = 115 X = 57 z = 0.5 z = 0.86 z = % 79.5% 89.44%

19 T Distribution When samples are small (less than 30), the distribution of their means might not be normally distributed Instead, they follow a slightly different distribution, called t-distribution Symmetrical like normal distribution but gets increasingly flat as sample size gets smaller

20 Confidence Levels and Intervals Assuming a truly unbiased and representative sample, it is still possible sample does not represent the parameters of the population, simply due to chance Any given sample is not likely to be exactly like the population Central limit theorem: Collecting many samples will produce variations from population But if you plot all the samples, they will show a normal distribution, and mean of the samples will be population mean (assuming large sample sizes)

21 Confidence Levels and Intervals So if you obtain a sample statistic, how do you know whether it truly represents the population parameter, or is simply a sample that varies by chance? Based on central limit theorem, you can state the statistic in terms of probability, in the form of confidence level and confidence interval

22 Confidence Levels and Intervals Confidence level or level of significance Sample deviates from population according to a normal distribution 99% of sample means lie within 3 standard deviations of population mean 95% of sample means lie within 2 (1.96, to be exact) standard deviations of population mean 68% (68.28%) of sample means lie within 1 standard deviation of population mean 95% level is most commonly used Higher confidence level used when costs of making incorrect decision based on stated results are high

23 Confidence Levels and Intervals Confidence interval Based upon confidence level and standard deviation Margin of error that you are confident (at stated confidence level) you won t be wrong when inferring or estimating the population mean from the sample mean At 99% confidence level: At 95% confidence level: At 68% confidence level: x x x ± 3SD ± 1.96SD ± 1SD

24 Confidence Levels and Intervals Example: Mean age of sample is 33.2, SD = 2 We are 99% confident population is between 27.2 and 39.2 We are 95% confident population is between 29.3 and 37.1 We are 68% confident population is between 31.2 and 35.2 As we raise the confidence level, we become more and more sure of less and less

25 Confidence Levels and Intervals Example: 45% of Americans believe in the Easter Bunny, with a margin of error of 3% Translation: we are 95% sure that between 42% and 48% of Americans believe in the Easter Bunny Mean = 45, SD = % = the parameter inferred in the population 95% = the confidence level (95% assumed since not stated) 3% = the confidence interval

26 Hypothesis Testing Formulate two competing hypotheses Null hypothesis: H 0 : µ 1 = µ 2 µ 1 and µ 2 are means of the population from which samples hypothetically have been obtained Always assumes no change or no difference between populations Innocent until proven guilty Sample means are drawn from same or identical populations Alternative hypothesis: H 1 : µ 1 not equal to µ 2 There is a difference between populations Sample means are drawn from different populations

27 Hypothesis Testing Approach is to use statistics to state whether the null hypothesis is supported by the data: 1. Formulate null hypothesis 2. Calculate test statistic 3. Determine probability (p value) of obtaining test statistic assuming hypothesis is true, usually by referring to a table of test-specific critical values 4. Compare p value to pre-determined significance level (usually.05) If p-value is smaller than significance level, can state that the data does not support null hypothesis (reject null hypothesis) If p-value is greater than significance level, cannot reject null hypothesis Note that rejecting null hypothesis does not prove that the null hypothesis is false, just that the probability of it being true is very small

28 Hypothesis Testing Errors of Statistical Testing Results are stated in probability - always a chance that when rejecting/not rejecting a null hypothesis the decision will be wrong Error can happen in two ways: Type I error: Null hypothesis H 0 is correct and we reject it Conclude there is a difference when there really isn t Can occur only in situations where null hypothesis is rejected Type II error: Null hypotheis H 0 is wrong but we fail to reject it Conclude there is no a difference when there really is Can occur only in situations where null hypothesis is not rejected

29 Parametric Versus Non-Parametric Methods Parametric statistical methods assume: Normal distribution When comparing groups, scores exhibit similar variance or spread Interval or ratio level data Non-parametric methods do not make assumptions about the sample of the population distribution Data are categories or ranks (nominal or ordinal) Usually less powerful Need larger samples

30 T Test Parametric test to determine whether there is a significant difference between two sample means Two types of t-tests: Two independent groups Scores in each group are unrelated Use independent t test Matched pairs or repeated measures Scores in groups are related Use paired t test More powerful than independent; higher probability of finding significant difference is one exists, because paired design reduces variability by eliminating individual differences

31 T Test Goal of t test: is there a difference in the populations based on data from samples of those populations? Based on two factors: Sample mean difference Difference between the two sample means Larger the mean difference, more likely there is a difference between the two populations Sample data variability Greater variability reduces likelihood that the sample mean difference is result of a real difference

32 T Test Example: The library is investigating two methods of teaching library skills to undergraduates. Twenty freshmen were randomly assigned to method COMPUTER and another twenty to method PAPER. After two weeks of instruction, the librarians computed the following from a library skills test. Method N Mean SD COMPUTER PAPER

33 T Test Hypothesis: H 0 : µ 1 = µ 2 Using formula, t statistic is 2.07 Need to determine probability that null hypothesis is true (p-value) based on t score t table T table requires degrees of freedom and significance level Degrees of freedom: N1 + N2-2 Significance level: 0.05 Example: df = = 38 Look up critical value of t in T table for df 30, t05 t critical value = t = 2.07 critical value of t = 2.024

34 T Test t = 2.07 critical value of t = Do we accept or reject null hypothesis? Is there a statistically signficant difference between the samples?

35 T Test t = 2.07 critical value of t = Do we accept or reject null hypothesis? Is there a statistically signficant difference between the samples? We reject null hypothesis because t is greater than the critical value We are confident at a 95% level that the difference in mean scores between methods is not due to chance Method N Mean SD COMPUTER PAPER

36 Chi-Square Test Basic purpose is similar to t test Less powerful than t test Why use? Wider applicability -- can deal with: Samples from data sets that are not normally distributed Interval and nominal data More than two samples One-sample X 2 test Two-sample X 2 test Three or greater sample X 2 tests

37 Chi-Square Test Performing chi-square test Contingency table Columns represent categories of one variable, rows categories of second variable Entries in the cells indicate the frequency of cases for a particular row Expected frequency is the theoretical frequency we would expect to see if the two variables were not related Calculate expected frequencies for each cell = (Row marginal total x column marginal total) / grand total Yes No Total Male 5 (15) 45 (35) 50 Female 25 (15) 25 (35) 50 Total

38 Chi-Square Test Null hypothesis assumes the variables are not related Test statistic chi-square score: X 2 Summarizes the discrepancies between the observed and the expected frequencies Determine the probability that the null hypothesis is true (the p-value) based on the test statistic Refer to table of critical values of chi-square, using df and significance level Larger the X 2 score, smaller the probability that the null hypothesis is true When the calculated X 2 score is equal to or greater than the critical value listed in the chi-square table, reject null hypothesis

39 Chi-Square Test Null hypothesis assumes the variables are not related Example: X 2 = Df = (#rows 1) X (#columns-1), so df = (2-1) x (2-1) Look up critical value for df 1, p =.05 in chi square critical values table Critical value = 3.84, X 2 = When the calculated X 2 score is equal to or greater than the critical value listed in the chi-square table, reject null hypothesis Yes No Total Male 5 (15) 45 (35) 50 Female 25 (15) 25 (35) 50 Total

40 Correlation Describes relationship between two sets of data Measurement of association Are variable X and variable Y related? How are variable X and variable Y related? How strongly are X and Y related?

41 Correlation Positive correlation Value of one variable increases as the value of other variable increases Height and weight Advertising and sales Book borrowing frequency and frequency of book repair Negative correlation Value of one variable decreases as the value of other variable increases Highway speeders and state trooper vehicles Recall and precision

42 Correlation Linear correlation: perfect correspondence between variables As X increases by constant quantity, Y increases by constant quantity Can be perfect positive (correlation coefficient = +1) or perfect negative correlation (correlation coefficient = -1) Curvilinear correlation: variables correlate along a curved line Age and coordination No correlation: when one variable changes, the other shows no trend to change in a uniform way (correlation coefficient = 0)

43 Correlation

44 Correlation Measuring strength of a relationship Calculate correlation coefficients Pearson s r r = z x z y N Derived from z scores of two distributions to be correlated Interval or ratio data, assumes normal distribution of variables and linear relationship between variables If assumptions are not met, can use version called Spearman s Rank Correlation Coefficient

45 Correlation Pearson r correlation coefficient ranges from -1 to 1 Positive r means positive correlation and negative r means negative correlation Closer r is to 0, weaker the relationship

46 Correlation Testing significance of Pearson r Pearson r coefficient by itself is a descriptive statistic To conclude that the r coefficient indicates a relationship in the population, must test for signficance of r

47 Correlation Testing significance of Pearson r Same basic procedure for hypothesis testing: Formulate null hypothesis Calculate test statistic Determine probability (p value) of obtaining test statistic assuming hypothesis is true Refer to table of critical values of r, based on degrees of freedom (number of pairs - 2) Compare p value to pre-determined significance level (usually.05) If p-value is smaller than significance level, can state that the data does not support null hypothesis (reject null hypothesis) If p-value is greater than significance level, cannot reject null hypothesis

48 Correlation Correlation and causation Does correlation mean causation? Statistically significant correlation coefficient only means that there is a strong chance of a real relationship between variables, not just coincidence or chance Does not mean that X caused Y or that Y caused X Other factors, or a third variable Z could actually affect both X and Y Even without causation, correlation is useful to know Can make informed prediction of value of one variable based on value of another

49 Regression Knowing two variables are correlated, how can we predict value of one variable based on the other? Regression analysis Variety of forms, most common is linear regression Goal is to find the best-fit straight line through data set Single straight line that comes closest to all of data points in scatter plot Y = a + bx

50 Regression Regression equation and regression line Least-squares regression Define a line such that the sum of the squared residuals are minimized Residuals are difference between actual and predicted values -- observed y-values and the y-values predicted by solving for x in the regression equation Using formula for least-squares, you end up with a regression equation that describes the best-fit line: y = x

51 Regression With regression equation, can input value for X and calculate the corresponding value for Y Calculating accuracy of prediction: coefficient of determination Square of correlation coefficient r: r 2 If r is.8, coefficient of determination r 2 is.64, which means 64% of variation in Y can be attributed to X

52 Regression Multiple regression Widely used research tool for anlayzing more complex situations Predicts value of variable based on 2 or more other variables

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