Statistics: Normal Distribution, Hypothesis Testing, and Chi-Square Test, Exams of Psychology

Download Statistics: Normal Distribution, Hypothesis Testing, and Chi-Square Test and more Exams Psychology in PDF only on Docsity! Statistics for Psychology Final Exam !!! Descriptive Statistics - summarizes a large amount of data Inferential Statistics - t the goal of inferential statistics is to use the limited information from samples to draw general conclusions about populations. The basic assumption of this process is that samples should be representative of the populations from which they come. This assumption poses a special problem for variability because samples consistently tend to be less variable than their populations. Median - middle value in a set of scores if there is an even number of scores, the average of the 2 middle scores is the median. Mean - the arithmetic average of a set of scores M= Σ X/N Mode - the most frequent value in a set of scores Standard Deviation - square root of variance, approximately the average amount that scores in a distribution vary from the mean =√Σ ((X-M)^2)/N = √SS/N = SD the approximately the average distance from the mean in a given data set. Variance - measure of how spread out a set of scores are; average of the squared deviation from the mean =Σ ((X-M)^2)/N = SS/N = SD^2 the average squared squared distance from the mean in a given data set. variable - a variable! you got this :) independent variable - the variable that is manipulated dependent variable - the variable that is measured to see if it is influenced by the independent variable Ceiling Effect - scores piled up at the top because there are no higher options ex: a test that is very easy would likely show a ceiling effect Floor Effect - scores piled up at the bottom because there are no lower options. ex: a test that is very hard would likely show a floor effect define types of variable ratio interval rank order nominal - 1.ratio --> interval scale with absolute zero point, ratio scales are subset of interval scales in whcih there is an absolute zero point, which means there is some value on the scale - usually zero - which means the complete absence of that variable. It is sometimes discreate, and sometimes continuous. proportion of a particular outcoe you would get if the experiment were repeated many tiems z-test for a single subject a.when to use 1.research and null hypothesis 2.characteristics of comparison dist 3.how to find sample cut off score 4.how to find sample score 5.how to reject or fail to reject null hypothesis - a. You use the z-test when you are comparing an individual to a sample. 1. H0= there is no difference between this individual and people in general H1= there is a difference between this indvidual and people in general 2. z-dist of individuals scores, µ=0 and σ=1.00 3. look at z-table for appropriate type of test (two-tail, or one tail), and sig level (alpha) to determine z- crit. If two tailed and 5% sig level, zcrit is +/- 1.96. if one tailed and 5% sig level, zcrit is +/- 1.64 4. z= (x-M)/SD 5. compare score in step 4 w/cut off level found in step 3 to decide whether or not to reject null hypothesis. Logic and foundation in testing hypothesis - 1. Assume no difference 2. Gather data that shows a difference 3. Conlude our assumption is incorrect 4. Infer that there is a difference 1.Reporting z-score in journal 2.Reporting z-score in plain english - 1.A difference between (insert a description of the population in terms of the dependent variable) and (insert a description of the sample in terms of the dependent variable) is or is not statistically significantly different, z = 0.00, p = .000. 2. We found a significant difference between blank and blank outlier - extreme scores compared to other values one tailed and two tailed hypothesis tests - two tailed --> hypothesis testing procedure for a non-directional hypothesis ex: H0= there is no difference between this individual and people in general H1= there is a difference between this indvidual and people in general one tailed -->hypothesis testing procedure for a directional hypothesis ex: we expect our observed individuals mean to be higher than the mean of people in general. we expect our observed indviduals mean to be lower than the mean of the people in general. Z-test for a sample when to use research and null hypothesis characteristics of comparison dist how to find sample cut off score how to find sample score how to reject or fail to reject null hypothesis - a. You use the Z-test when you are comparing a sample to a population with a known mean and standard deviation 1. H0= A sample who performs a certain activity have the same scores as the general population H1= A sample who performs a certain activity have different scores than the general population 2. distribution of means, µm, σm and will be normal if original dist is near-normal or N>30 ,- µm=µ - σ2m = σ2/N - σm = √σ2m 3. look at Z-table for appropriate type of test (two-tail, or one tail), and sig level (alpha) to determine z- crit. If two tailed and 5% sig level, zcrit is +/- 1.96. if one tailed and 5% sig level, zcrit is +/- 1.64 4. Z= (M - µm)/ σm 5. compare score in step 4 w/cut off level found in step 3 to decide whether or not to reject null hypothesis. 1.Reporting Z-score in journal 2.Reporting Z-score in plain english - same as z-score above ^ Distribution of means - Collection of means of all possible samples of a given size from a population. (also called sampling distribution of the means) Central Limit Theorem - For a population with mean σ and variance σ2: 1. Mean of distribution of means: µm=µ 2. Variance of distribution of means: σ2m = σ2/N Standard deviation of dist. of means : σm = √σ2 3. . Distribution of means will be normal if: Original distribution is near-normal OR N > 30 sampling error - discrepency between a given sample mean and the population mean 4. t= (M - µ)/ Sm 5. compare score in step 4 w/cut off level found in step 3 to decide whether or not to reject null hypothesis. Reporting single sample t-test in journal Reporting single-sample t-test in plain english - 1. Our sample of Bard students had more Facebook friends (M=248.11, SD=190.26), than the population (µ=200), t(71[df])=2.15, p<0.05. 2.Our sample of bard students had more facebook friends than the population. Biased Estimate - Estimate of population parameter, based on sample data that systematically underestimates of overestimates the true population parameter. This bias is in the direction of underestimating the population value rather than being right on the mark. Fortunately, the bias in sample variability is consistent and predictable, which means it can be corrected. For example, if the speedometer in your car consistently shows speeds that are 5 mph slower than you are actually going, it does not mean that the speedometer is useless. It simply means that you must make an adjustment to the speedometer reading to get an accurate speed. In the same way, we will make an adjustment in the calculation of sample variance. The purpose of the adjustment is to make the resulting value for sample variance an accurate and unbiased representative of the population variance. Unbiased estimate of population variance - Estimate of a population parameter, based on sample data, that is equally likely to underestimate or overestimate the true population parameter. using n-1 gives us an unbiased estimate S^2 = Σ(X-M)^2)/N-1 = SS/N-1 unbiased estimate of populations SD - S=√S^2 Degree of freedom - number of scores free to vary when estimating a populations parameter S^2 = Σ(X-M)^2)/N-1 = SS/N-1 = SS/df df = N-1 Symbol Check single sample t-test 1. S^2 2. S 3. S^2m 4. Sm - 1. S^2 --> estimated population variance 2. S --> estimated population SD 3. S^2m -->variance of t-distribution based on estimated population variance 4. Sm --> SD of t-distribution ( based on estimated population variance) T- statistic - The t statistic is used to test hypotheses about an unknown population mean when the value of is unknown. The formula for the t statistic has the same structure as the z- score formula, except that the t statistic uses the estimated standard error in the denominator. t-test for dependent means when to use research and null hypothesis characteristics of comparison dist how to find sample cut off score how to find sample score how to reject or fail to reject null hypothesis - a. You use the t-test for dependent means when you are comparing two means that are dependent on one another; population variance unknown. Comparing difference scores to some value (usually zero) 1. H0= There are no differences between the means of the two related groups H1= There are differences between the means of the two related groups 2. t-distribution with N-1 df, μ= 0, Sm based on estimated population variance (S2) 3. look at t-table for appropriate type of test (two-tail, or one tail), and sig level (alpha), and look at df to determine t-crit. 4. t= (M - µ)/ Sm 5. compare score in step 4 w/cut off level found in step 3 to decide whether or not to reject null hypothesis. 1.Reporting t-test for dependent means in journal 2.Reporting t-test for dependent means in plain english - 1.We reject the null hypothesis that there is no difference between the means of the two related groups 2.ex. Women took longer to answer questions about sentences related to angry content after Botox (M = 8.40) than before Botox (M = 6.60), t (4) = 3.09, p <.05. t-test for independent means when to use research and null hypothesis characteristics of comparison dist how to find sample cut off score how to find sample score how to reject or fail to reject null hypothesis - a. Comparing two separate groups; population variance unknown 1. H0= The population means from the two unrelated groups are equal H1= The population means from the two unrelated groups are not equal 2.Distribution of difference between the means: t-distribution with N-2 df, µdiff = 0 -the risk we are willing to take that we will reject a true null if alpha is changed from .05 to .01 then.. -type I error decreases -type II error increases Beta - Power + β = 1 the probability of Type II error in any hypothesis test-incorrectly concluding no statistical significance. (1 - Beta is power). Regression coefficients. In most textbooks and software packages, the population regression coefficients are denoted by beta. effect size 1.equation for single sample and dependent t-test 2.equation for independent t-test - Effect size standardized measure of distance between populations (degree to which populations overlap) Large --> 0.80 Medium --> 0.50 Small ---> 0.20 1. d= μ1 - μ2/σ 2. d= M1-M2/Spooled Cohens d - Cohen's d is an effect size used to indicate the standardised difference between two means. It can be used, for example, to accompany reporting of t-test and ANOVA results. It is also widely used in meta-analysis. Cohen's d is an appropriate effect size for the comparison between two means. meta analysis - statistical method for combining effect sizes from different studies. one-way Analysis of Variance (ANOVA) when to use research and null hypothesis characteristics of comparison dist how to find sample cut off score how to find sample score how to reject or fail to reject null hypothesis - a. You use a one-way ANOVA when you are Comparing more than two independent groups on an interval or ratio variable 1. H0= means are all the same. "No difference" u1=u2=u3=u4 H1= At least one mean is different from the others do not know which one is different, we can just predict that a difference does exist 2. F-distribution. Comparison distribution for an ANOVA based on...Degrees of freedom between. df used in the between-groups estimate of population variance (Ngroups-1) and Degrees of freedom within. df used in the within-groups estimate of population variance (df1 + df2 + df3 + ... + dflast) F-distribution: dfbetween = Ngroups -1 dfwithin = df1+df2+...+dflast 3. Look up value for correct dfbetween, dfwithin, α 4.F= S^2between/S^2within = MSbetween/MSwithin 5.Compare F (dfbetween, dfwithin) to Fcritical (dfbetween, dfwithin). Make decision. Reporting one-way ANOVA journal Reporting one-way ANOVA in plain english - 1. An analysis of variance showed that the effect of type of variable 1 on variable 2 was significant, F (__,__) = ____, p = ____ , n^2 (effectsize) = 2.A One-way ANOVA was conducted to compare effect of [name the effect (IV)] on the (dependent variable ANOVA table - Sum of Squares -SSwithin -SSbetween -SStotal = SSbetween + SSwithin Df -dfwithin -dfbetween -dftotal = N - 1 = dfbetween + dfwithin Mean Square -SS/df F -MSbetween/MSwithin 1.Within-group variance 2. MSwithin - 1.variability based on individuals within groups. Factors common across all groups 2.FIGURING OUT WITHIN-GROUP ESTIMATE OF VARIABILITY (MSwithin) A pooled estimate of population variance MSwithin= (S1^2 + S2^2 + .... Sn^2)/Ngroups 1.Between-group variance 2. MSbetween - 1.variability based on our variable of interest distance between group means. 2. FIGURING OUT BETWEEN-GROUP ESTIMATE OF VARIABILITY S2between= S2m n∙ Degrees of Freedom Between - Independent variable with at least two levels]. 3rd Null Hypothesis - The Interaction Effect There is no significant interaction effect between the [Insert the 1st Independent variable] and the [Insert the 1st Independent variable] in terms of the [Insert the Dependent Variable]. 2. F-distribution. Comparison distribution for an ANOVA based on...Degrees of freedom between. df used in the between-groups estimate of population variance (Ngroups-1) and Degrees of freedom within. df used in the within-groups estimate of population variance (df1 + df2 + df3 + ... + dflast) F-distribution: dfbetween = Ngroups -1 dfwithin = df1+df2+...+dflast 3. Look up value for correct dfbetween, dfwithin, α 4.F= S^2between/S^2within = MSbetween/MSwithin 5.Compare F (dfbetween, dfwithin) to Fcritical (dfbetween, dfwithin). Make decision. Reporting factorial ANOVA journal Reporting factorial ANOVA in plain english - 1.A two-way analysis of variance was conducted on the influence of two independent variables (athlete type, age) on the number of slices of pizza eaten in one sitting. Athlete type included three levels (football, basketball, soccer players) and age consisted of two levels (younger, older). All effects were statistically significant at the .05 significance level except for the Age factor. The main effect for athlete type yielded an F ratio of F(2, 63) = 136.2, p < .001, indicating a significant difference between football players (M = 9.39, SD = 1.99), basketball players (M = 5.17, SD = 1.40) and soccer players (M = 2.52, SD = 1.53. The main effect for age yielded an F ratio of F(1, 63) = 2.9, p > .05, indicating that the effect for age was not significant, younger (M = 5.97, SD = 3.97) and older (M = 5.39, SD = 2.34) The interaction effect was significant, F(2, 63) = 13.36, p < .001. 2.A Factorial ANOVA was conducted to compare the main effects of [name the main effects (IVs)] and the interaction effect between (name the interaction effect) on (dependent variable) interaction - Effect of a combination of variables that wouldn't be seen from any of the variables individually. main effect - Results for one of the variables, averaging across levels of other variables. ANOVA - effect size - Effect size in ANOVA = proportion of variance accounted for effect size equation = n^2 = SSbetween/SStotal effect size sizes small --> 0.01 medium --> 0.06 large --> 0.14 As effect size increases, power increases. Correlation when to use research and null hypothesis characteristics of comparison dist how to find sample cut off score how to find sample score how to reject or fail to reject null hypothesis - a. you use a correlation test when you are examining the relationship between two variables (USUALLY two interval or ratio variables)Correlation measures the relationship between two variable. Reflects how much variables move away from the mean in the same (or a different) direction 1. Two-tailed - non- directional H0= There is no relationship between variable 1 and variable 2 r = 0 H1= There is a relationship between variable 1 and variable 2 r ≠ 0 2. SPSS 3. SPSS 4.SPSS 5.SPSS problems in interpretation of correlation tests - Direction of causality Restriction in range Nonlinear relationships Unreliable measures Outliers scatterplot - a visual method used to display a relationship between two interval-ratio variables -often used as a first exploratory step in regression analysis -can suggest whether two variables are associated -can suggest the existence, strength, direction and linearity of two variables properties of correlation (3) - 1.Range between -1.0 to +1.0 2.The sign indicates the direction of the relationship 3.The absolute value indicates the strength of the relationship (higher -> stronger relationship; lower -> weaker relationship; 0 = no relationship) Notation Check for Correlation r^2 - r^2 is % of shared variance R^2is the percent of variance account for by our predcitor(s) used for correlation and regression proportionate reduction in error in multiple regression & analysis of variance effect size for correlation - r is itself a measure of effect size! Reporting regression result in plain english - 1.Social support significantly predicted depression scores, b = -.34, t(225) = 6.53, p < .001. Social support also explained a significant proportion of variance in depression scores, R2 = .12, F(1, 225) = 42.64, p < .001. 2. social support significantly predicted depression scores. social support also explained a significant proprortion of variance in depression scores. linear relationship - a relationship between two interval-ratio variables in which the observations displayed in a scatter diagram can be approximated with a straight line multiple regression - Trying to predict a person's score on one variable from his or her scores on multiple variables equation for regression - ^ Y = Y'= bX +a Y= the predicted score on the DV X= the score on the IV a= Y-intercept b= slope Bivariate regression - the analysis of relationships between two interval-ratio variables using a regression equation model *1 IV and 1 DV regression coefficient - Beta (β) is thestandardized regression coefficient regression coefficient in standard deviation units -Y = Y'= bX +a -> b is our Regression Coefficient For every unit increase in our predictor variable X, our criterion variable increases by b Regression line Why do we use the regression line? - in real research we rarely find a perfect linear relationship so when the DV is not completely determined by the IV not all the observations (dots) will lie exactly on the line; thus the best fitting line is the one that generates the least amount of error this is the best line (the one that minimizes error) (the best fitting line or least squares line) why do we use the regression line? 1) to see the trends 2) To predict the value of Y(DV) for any change in the X(IV) problems in interpretation with regression - Direction of causality Restriction in range Nonlinear relationships Unreliable measures Outliers 1. predictor variable 2. criterion variable 3. what is their relationship? - 1.variable used to predict an individual's score on another variable 2.a variable that is predicted 3.For every 1 standard deviation increase in your predictor variable, you expect a 1 standard deviation increase in your criterion variable correlation vs. regression - correlation= association regression= prediction of DV (Y) based on the IV (X) Chi-Square goodness of fit when to use research and null hypothesis characteristics of comparison dist how to find sample cut off score how to find sample score how to reject or fail to reject null hypothesis assumptions - a. You use a chi-square goodness of fit test when you are Examining observed frequency of a single nominal variable, Examines how well an observed frequency distribution of a nominal variable fits expected frequencies.Compares how well a single nominal variable's distribution fits an expected distribution 1.H0=Distribution in our sample is the same as in the population. H1=Distribution in our sample is NOT the same as in the population. 2. χ2 distribution with df = Ngroups-1 3. Look at table with appropriate α and df 4. X2=Σ(O-E)^2/E -- you might need to calculate expected! 5.Compare your score in Step 4 with cutoff level in Step 3 Reporting Chi-Square goodness of fit test result in journal Reporting Chi-Square goodness of fit test result in plain english -