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Jump to first page Inferring Sample Findings to the Population and Testing for Differences.

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Presentation on theme: "Jump to first page Inferring Sample Findings to the Population and Testing for Differences."— Presentation transcript:

1 Jump to first page Inferring Sample Findings to the Population and Testing for Differences

2 Statistics versus Parameters statistics! n Values computed from samples are statistics! parameters! n Values computed from the population are parameters! n Use Greek letters when referring to parameters. n Use Roman letters for statistics.

3 Inference and Statistical Inference n Inference n Inference - generalize about an entire class based on what you have observed about a small set of members of that class. n Draw a conclusion from a small amount of evidence. n Statistical Inference n Statistical Inference - Sample size and sample statistics are used to make estimates of population parameters.

4 Hypothesis Testing n Statistical procedure used to accept or reject the hypothesis based on sample information. n Steps in hypothesis testing u Begin with a statement about what you believe exists in the population. u Draw a random sample and determine the sample statistic. u Compare the statistic with the hypothesized parameter. u Decide whether or not the sample supports the original hypothesis u If the sample does not support the hypothesis, revise the hypothesis to be consistent with the sample's statistic.

5 Test of the Hypothesized Population Parameter Value The sample mean is compared to the hypothesized mean, if z exceeds critical value of z (e.g., 1.96) then we reject the hypothesis that the population mean is Mu. For example, we hypothesize that the average GPA for business majors is not the same as Recreation majors.

6 Directional Hypotheses n Indicates the direction in which you believe the population parameter falls. n For example, the average GPA of business majors is higher than the average GPA of Recreation Majors. n Note that we are now interested in the volume of the curve on only one side of the mean.

7 Interpretation n If the hypothesis about the population parameter is correct or true,then a high percentage of the sample means must fall close to this value (i.e., within +/-1.96 sd.) n Failure to support the hypothesis tells the hypothesizer that the assumptions about the population are in error.

8 Testing for Differences Between Two Means n H o : There is no difference between two means. (Mu 1 =Mu 2 ) n H a : There is a difference between two means. (Mu 1  does not equal Mu 2 ).

9 Testing for Differences Between Two Means: Example n Is there a statistically significant difference between men and women on how many movies they have seen in the last month? n H o : There is no difference between two means. (Mu W =Mu M ) n H a : There is a difference between two means. (Mu W /=Mu M )

10 GenderNMeanSt. Dev male192.36841.98 female132.53852.18 tdfSignificance (2 tailed) TOTAL-.22930.820 FSig. Levene’s test for equality of variance.004.952 Example

11 Testing for Differences Between Two Means n Fail to reject the null hypothesis, that the means are equal n Why? u Significance =.82 u Reject any significance lower than.05 u.82 >,05; therefore, fail to reject null u there is no statistically significant difference between men and women on how many movies seen in last month u makes sense - look at means (2.36 & 2.53)

12 Small Sample Size - t-Test n Normal bell curve assumptions are invalid when sample sizes are 30 or less. n Alternative choice is t-Test n Shape of t distribution is determined by sample size (i.e., degrees of freedom). n df = n-1

13 ANOVA n ANOVA = Analysis of Variance n Compares means across multiple groups n ANOVA will tell you that one pair of means has a statistically significant difference but not which one n Assumptions: u independence u normality u equality of variance (Levene test)

14 Analysis of Variance When researchers want to compare the means of three or more groups. ANOVA used to facilitate comparison! Basic Analysis Does a statistical significance difference exist between at least two groups of means? ANOVA does not communicate how many pairs of means are statistically significant in their differences.

15 Hypothesis Testing H o : There is no difference among the population means for the various groups. H a : At least two groups have different population means.. When MS Between is significantly greater than MS Within then we reject H o.

16 F value F = MS Between /MS Within If F exceeds Critical F (df1, df2) then we reject H o.

17 Visual Representation Population 1 Population 2 Population 4 Population 5 Population 3 Appears that at least 2 populations have different means.

18 Visual Representation Population 1 Population 2 Population 4 Population 5 Population 3 Appears that populations do not have significantly different means.

19 Tests of Differences n Chi-square goodness-of-fit u Does some observed pattern of frequencies correspond to an expected pattern? n Z-test/T-test u Is there a significant difference between the means of two groups? n ANOVA u Is there a significant difference between the means of more than two groups?

20 When to Use each Test n Chi-square goodness-of-fit u Both variables are categorical/nominal. n T-test u One variable is continuous; the other is categorical with two groups/categories. n ANOVA u One variable is continuous (i.e., interval or ratio); the other is categorical with more than two groups.

21 How to Interpret a Significant p-value (p <.05) n Chi-square goodness-of-fit u “There is a significant difference in frequency of responses among the different groups (or categories).” n T-test u “The means (averages) of the 2 population groups are different on the characteristic being tested.” n ANOVA u “The means of the (multiple) population groups are different - need post hoc test (e.g., Bonferroni) to determine exactly which group means are different from one another.”

22 Measuring Association n Is there any association (correlation) between two or more variables? n If so, what is the strength and direction of the correlation? n Can we predict one variable (dependent variable) based on its association with other variables (independent variables)?

23 Correlation Analysis n estimate of correlation between two A statistical technique used to measure the closeness of the linear relationship between two or more variables. n Can offer evidence of causality, but is not enough to establish causality by itself (must also have evidence of knowledge/theory and correct sequence of variables). n Scatterplots can give visual variables.

24 Regression Analysis n Simple Regression u relate a single criterion (dependent) variable to a single predictor (independent) variable n Multiple Regression u relate a single criterion variable to multiple predictor variables  All variables should be at least interval!

25 Correlation/Regression n Coefficient of Correlation (r) u measure of the strength of linear association between two variables u also called “Pearson’s r” or “product-moment” u ranges from -1 to +1 n Coefficient of Determination (r 2 ) u proportion of variance in the criterion explained by the fitted regression equation (of predictors)


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