Posted: May 12th, 2022

R7031 M6 Multiple Regression – Assignment 1: Discussion Question,Assignment 2: Quiz

Assignment 1: Discussion QuestionThis module covered the following topics:The Language of Association and PredictionPearson CorrelationThe Correlation MatrixSimple RegressionMultiple RegressionBy Friday, February 20, 2015, please post the questions you have about any of the above topics to the Discussion Area. Identify the specific topic and ask a clear, succinct question. Indicate where in the module, text, or WebEx demo your question occurs. This includes technical terms, equations, results, or interpretations.Then, review your fellow students questions. Is there anything you can clarify for them? Are you also confused about the same topic? Post your comment to at least two other students by Wednesday, February 25, 2015. (The instructor will also expand and clarify.)All written assignments and responses should follow APA rules for attributing sources.Assignment 1 Grading CriteriaPresented a clear and thoughtful question or comment.Used vocabulary relevant to the current module’s topics.Participated in the discussion by asking a question, providing a statement of clarification, providing a point of view with rationale, challenging a point of discussion, or making a relationship between one or more points of the discussion.Justified ideas and responses by using appropriate examples and references from texts, Web sites, and otherreferences or personal experience.___________Unit 6 : Module 6 – M6 Assignment 2 QuizAccess dates:Can be reviewed in Gradebook on:Number of times this exam can be taken:Time allowed to complete:2/19/2015 12:00:00 AM to 2/25/2015 11:59:00 PM2/25/2015 12:00:00 AM11hAssignment 2: QuizDue Sunday, February 22, 2015.For each of the research questions below, identify thevariables and the method of analysis to best answer thequestion. The best method of analysis may be materialcovered in Modules 3, 4, 5, or 6.Present your answer with the following template where IV=Independent Variable(s) and DV= Dependent Variable(s).For each Variable identify if it is continuous orcategorical/dichotomous to help you identify the bestmethod_IV=DV=Covarite=Best method of analysis=Assignment 2 Grading CriteriaMaximumPointsAnswered all 10 questions accurately: (6pts/question)Total:Identified the IV , DV , Covariate,or designated it as not applicablen/a for each question (3 pts)60Identified the best method ofanalysis for each question (3 pts)60____________________________________________________________________________Assignment 3: Application1. Go to Doc Sharing and choose the database and assignment file you plan towork with for this module. It is highly recommended that you continue to use this filethroughout the rest of the course.Name of DatabaseName of Assignment FileR7031business.savR7031business.savassignments.docR7031counseling.savR7031counseling.savassignments.docR7031education.savR7031education.savassignments.docFollow the instructions for the Module 6 Assignment. There are threequestions to answer.Your assignment will be submitted as a Microsoft Word document. Transferappropriate SPSS tables (see note below) into your document to support yourconclusions.Name your assignment R7031M6yourlastname.doc and submit it to the M6:Assignment 3 Dropbox by Tuesday, February 24, 2015.NOTE:1. In SPSS, set your tables to APA Style.2. From the data window (spreadsheet), select Edit and then Options from themenu.3. In the options Menu, click on the Pivot Tables folder.4. Select the academic style and then click OK. All tables will now be producedusing the selected format.5. Bring your tables into Microsoft Word by:___________________________________________________________________Calculate your statistic and p-valueASSUMPTIONS.Let’s first check to see if the assumptions have been met.Normal Distribution of the variables: Examining the measures of central tendency and variability, we see that all of the variables have not violated this assumption: the means, medians and modes are “close,” and the indicators of skewness and kurtosis are well within the normal range (close to 0.00).AgeAverage number of hours of sleep during weekN Valid4040Missing00Mean44.806.86Median42.006.75Mode41.006.00Std. Deviation10.431.275Skewness.710.411Std. Error of Skewness.374.374Kurtosis-.247-.131Std. Error of Kurtosis.733.733Minimum30.004.00Maximum69.0010.00LinearityNext, we look at the scatterplots to see if there is enough of a linear relationship. We visualize an “ellipse” around the dots.Homoscedasticity—the Distribution of ResidualsWe examine the plot of the residuals as compared to the predicted values, The lack of a pattern (i.e., the dots are scattered randomly) supports the assumption of homoscedasticity (equal variance of error across values of the independent variables).StatisticsLet’s take a look at the statistics that will answer our research questions.Have we explained a large portion of variance?Model Summary(b)ModelRR SquareAdjusted R SquareStd. Error of the Estimate1.557.310.2921.07a Predictors: (Constant), Ageb Dependent Variable: Average number of hours of sleep during weekThere is R, which is the measure of association—in this case, the same as the Pearson r = -.557. While the Pearson’s r coefficient indicates direction (positive or negative), the same statistic in regression is only looking at strength and not direction. That explains why the R statistic in the Model Summary (b) does not have a negative sign like the Pearson’s r.R2 = .310, or 31% of the variance is explained—69% is not. The adjusted R2reduces the R2 by taking into account the sample size and the number of independent variables in the regression model (it becomes smaller as we have fewer observations per independent variable). The Standard Error of the Estimate is the standard deviation of the distribution of residuals.Is this amount of variance accounted for significant? Look at the ANOVA SOURCE TABLE.ModelSum of SquaresdfMean SquareFSig.Regression19.701119.70117.0950.000Residual43.793381.152Total63.49439a Predictors: (Constant), Ageb Dependent Variable: Average number of hours of sleep during weekThe F-test(the ratio of regression to residual variation) is F(1, 38) = 17.095, p<001. The model is statistically significant.And last, we examine the regression coefficient. In the case of one predictor, ? (Beta) is the same as the correlation coefficient, -.557. So we can say this indicates a moderate negative impact, that is statistically significant, at t = -4.135, p<.001.Coefficients(a)ModelUnstandardized CoefficientsStandardized CoefficientstSig.BStd. ErrorBeta1(Constant)9.9150.75813.0870.000Age-0.0680.016-0.557-4.1350.00Dependent Variable: Average number of hours of sleep during weekRetain/reject the null hypothesisWe have two research questions and two hypotheses.1. Does Age predict sleep?H0: R2 = 0H1: R2>0R2 = .310, and the F-test (the ratio of regression to residual variation is F(1, 38) = 17.095, p<.001. The model is statistically significant.We reject the null hypothesis.2. How good a predictor is Age?H0: B = 0H1: B ? 0The standardized coefficient ? = -.557, indicate a moderate negative impact, t = -4.135, p <.001.We reject the null hypothesis.Risk of Type I and Type II errorType I Error R2 is a measure of effect size, and the interpretation using the effect size chart from previous lessons can be applied here. Notice that it is similar to the discussion of size and strength.Value of dHow strong the effect0.00 to .20No effect to small effect.21 to .33Small to moderate effect.34 to .50Moderately strong effect.51 to .75Strong effect.76 or moreVery strong effectRecall too that when R2 = .310, only 31% of the variance is explained, 69% is unexplained. While it is statistically significant, this should be interpreted cautiously.Type II Error Prediction question using survey methods for data collection are notoriously at risk for Type II error because of the consequences of unreliability on measuring the strength and direction of the predictive relationship. In our case, we are relying on participants to be honest and accurate about their sleep and their age! Unreliable measures can obscure the strength of relationship so that you may erroneously accept the null hypothesis.Reliability can be enhanced during the design and data collection process.State your results in APA style and formatIn writing this up for his dissertation, Mr. Shelby would state:A Simple Regression was used to examine age as a predictor of sleep. Results suggest age is a moderate predictor of sleep, and the model is statistically significant, R2 = .31, F(1, 38) = 17.10, p<.001.However, 69% of the variance remains unexplained. The standardized coefficient ? = -.56, indicates amoderate negative impact, t = -4.14, p < .001, indicating as age increases, sleep decreases.Go to Doc Sharing to download the M7 example database found in R7031 Example databases student zip file.Go to Doc Sharing to download the WebEx tutorial links. View the R7031 M7 Simple Regression tutorial.Multiple RegressionMoving From Simple to Multiple RegressionRecall the example from simple regression:“A student mentioned she has a hard time staying awake in class because she is having trouble sleeping. Mr. Shelby wonders if this is true for most students. The class calculates the mean hours of weeknight sleep. Then, a woman in her early fifties arrives late, and the instructor asks the class, 'Let’s predict how much weeknight sleep she gets.'"The choices are:PredictionMethodWorstGuess.BetterUse the mean.Even BetterUse a variable (like age) to improve upon the mean as a predictor.The BestUse ANOTHER variable in addition to age (like coffee consumption) to improve prediction even more.Multiple regression involves using two or more independent variables to improve prediction, i.e., to explain more variance in the dependent variable. Using the variance pie diagram, we can see that in the case of one predictor, we can “explain” some part of the variance in the dependent variable, as indicated where the circles overlap (green). But there’s a lot left in Y that remains “unexplained” (yellow).Figure M7.14.With more than one variable, the amount of unexplained variance is reduced, and our ability to predict becomes more accurate. Figure M7.14 shows that there is more explained variance in Y and much less “unexplained” variance in Y (yellow) when using threepredictors.So the formula for the line to be solved looks like this:Y = a + b1X1 + b2X2 + … bnXnThe “…BnXn” indicates that you can keep on adding variables.In the case of Mr. Shelby, recall—He is interested in quality of life of working adults in graduate school. He collected data on age, sleep patterns, and caffeine consumption from 40 women who were about to take their comp exams. He knows from reading the published literature that that sleep patterns are influenced by age and anxiousness.So, he could create a predictive question with many predictor variables, and ask two research questions.Can we predict average weeknight sleep from a combination of variables—in this case age, and anxiety?Which variables are the “best” predictors?In education, this kind of question is often used when asking if students’ self-perceptions have an impact on performance. So the research question might be:Do perceptions of self-efficacy AND perceptions of belonging predict academic performance?In organizational leadership, this kind of question is often used when asking what leadership characteristics best predict profitability. So the research question might be:Does tenure of leadership AND communication effectiveness predict profitability?In counseling, this kind of question is often used when asking what variables are most important in predicting the severity of depression. The research question might be:Does number of previous admissions AND client age predict the severity of the current depressive episode?Multiple Regression: MethodsMethods for Multiple RegressionImagine you are making a stew, and you have a pile of vegetables to put in the pot. Some cook quickly, some take a long time to cook.You can use several strategies to add the veggies:Add them all in at once.Add them in separate groups of veggies according to their cooking time.Let an expert cook (other than you) add them in according to their specifications.Some combination of the above.Adding variables into a multiple regression equation offers the same kinds of choices. Each choice has a different name and a different rationale.Name of the TechniqueWhat it doesAdvantagesSimultaneous (called “Enter” in SPSS)(1. Add them all in at once)Adds all the variables in at once—simultaneouslyIf you are primarily interested in how much variance is explained and don’t care about order or which variables are important.Hierarchical (called “Enter” + “Next” in SPSS)(2. Add them in separate groups according to their cooking time)Adds variables in separate groups you designateAllows you to examine the impact of groups of variables in terms of how each group increases explained variance. Also good if you have a hypothesis about the temporal sequence of the variables.Stepwise(3. Let an expert cook [other than you] and add them in one at a time)The computer determines what variables to add in—OR take out—on the basis of maximizing explained varianceAllows you to see at each “step” how much unique variance each variable contributes. Also good if you are exploring the best combination of predictors and you do not have a specific hypothesis about sequence.Enter, Stepwise, Forward, Backward, Remove(4. Some Combination of the Above)You alone or you in combination with the computer determine the order of entry.If you have a hypothesis you are testing, this gives you total control over the process.A Word about Unique VarianceA variable is considered a “good” predictor when it explains variance in Y (i.e., there is a lot of overlap in the X1-Y circle), but is independent of (i.e., minimum overlap) with other predictors (i.e., X2 circle).In the first picture, see that predictors X1 and X2 explain UNIQUE variance in Y, and have no SHARED variance.In the second picture, notice that X1 and X2 overlap with each other as much as with Y. Statistically, it means they are correlated with each other as much as with the Y variable.Statistically, the addition of variable X2 explains only a very small part of unique variance (the orange bit), while X1 explains much more unique variance (the green). So it is likely X2 will not show up in the final solution. It’s like adding red potatoes and yellow potatoes to your stew: you get a little extra color, but no added impact on flavor!Multiple Regression: StatisticsStatistics for Multiple RegressionWe will focus on the stepwise procedure for this example. Since Mr. Shelby doesn’t have a hypothesis about which variables should go first, this is an appropriate technique to choose.The statistics used to interpret multiple regression analysis are much the same as simple regression—only more of them, and one for each variable or variable group you add.SPSS makes the choice of what to include based on the variable that will explain the most UNIQUE variance. Once that variable is “in” the equation, SPSS looks at the remaining variables and goes through the choosing process until either all the variables are in the final module or the ones left out do not account for sufficient unique variance to be included.The statistics include:R2 for each time a variable is entered. SPSS uses the term “model” to denote the unique event of a variable being added or taken away.An F statistic to interpret the significance of the model each time a variable is added.A table of the “leftovers”—the excluded variables that weren’t included, and a statistic that tells us why.The source table for multiple regression analysis should look familiar. Model 1 corresponds to the first variable that is entered. Model 2 corresponds to the second variable that is entered.ModelSum of SquaresdfMean SquareFSig.1 RegressionResidualTotalSSregSSresSStotNo. of PredictorsN-2N-1SSregdf SSregdfMSregMSres2 RegressionResidualTotalSSregSSresSStotNo. of PredictorsN-3N-2SSregdf SSregdfMSregMSresNotice that for every variable entered, a model is generated, and each one produces an F-Test. That way you can see if significance increases with each variable added.We also examine the regression coefficients. Since we have more than one variable, the regression coefficients become useful.The unstandardized coefficients (B) are used to create the formula for the line.The standardizedcoefficients (?) are compared to see the size and direction (positive or negative).Size and direction tell us which of the predictors are most important.So the results of stepwise regression analysis can tell us: 1. How much variance is explained—R2 AND how much R2 goes up each time we add a predictor.2. Whether the amount of variance is statistically significant for each model.3. The importance of each predictor in terms of the size and direction of the standardized regression coefficient.4. Why some variables were included and others excluded..jpg" alt="http://myeclassonline.com/ec/courses/AUO_files/AU_img.gif">Multiple Regression: Assumptions.png” alt=”http://myeclassonline.com/ec/courses/AUO_files/AU_spacer.gif”>AssumptionsMultiple regression analysis builds on the three important assumptions we reviewed in simple regression and adds two more that should be checked before interpreting the results.Both X and Y variables must be close to a normal distribution. You can check this by examining the measures of central tendency, variability, skew, and kurtosis. This is called the normality of the variables assumption.The X and Y variables must form a linear relationship. That is, the scatterplot of X and Y must approximate a straight line, not a curved line. This is called the linearity assumption.The distribution of the residuals must be even about the same for all predicted scores. This is called the homoscedasticity assumption. In other words, you cannot have a lot “error” at lower ranges of Y and a little at the top ranges. The amount of error from low to high values of Y needs to approximate the normal distribution.The errors of the dependent variables are normally distributed. This is called the normality of the residual assumption.The independent variables are uncorrelated with each other (there is a minimum of overlapping circles among the predictors). This is calledmulticolinearity.SPSS provides scatterplots and statistics to assess the extent to which assumptions have been violated (or not).These assumptions are very important, for each time you add in a new variable, you are adding in “error”—unwanted variance. This could increase the risk of Type I error.Multiple Regression: Testing (1 of 3)Review of Hypothesis Testing ModelBack to our example. The soon-to-be Dr. Shelby was interested in the variables that could impact the health of working women attending graduate school. He’s considering how age and self-assessment of anxiety can impact regular weekday sleeping.1. State the hypothesisWe have two research questions and two hypotheses.Does the combination of independent variables predict Average Weeknight Sleep?H0: R2 = 0H1: R2>0Which independent variables are the best predictors?H0: B = 0H1: B ? 02. State your ?levelFor this question, alpha is set at .05.3. Collect the dataGo to Doc Sharing to download the M7 Example Database found in the R7031 Example databases student zip file.4. Calculate your statistic and p-valueASSUMPTIONS. First we check these.1. NORMALITY OF THE VARIABLES.We examine the descriptive statistics to verify normality of each of our variables. Do these look normal?AgeSelf-rating on Anxiety ScaleAverage number of hours of sleep during weekNValid404040Missing000Mean44.805.906.862Median42.00006.00006.750Mode41.006.006.00Std. Deviation10.432.0981.276Skewness.710-.019.411Std. Error of Skewness.374.374.374Kurtosis-.247-.200-.131Std. Error of Kurtosis.733.733.733Minimum30.001.004.00Maximum69.0010.0010.00Using the navigation on the left, please proceed to the next page.Multiple Regression: Testing (2 of 3)2. LINEARITY. SPSS produces a scatterplot for each of the variables used in the final analysis. These are partial correlation plots, meaning it displays the UNIQUE relationship between each independent variable and the dependent variable. You can see that both variables have a linear relationship with the dependent variable—both negative.3. HOMOSCEDASTICITY. Here we examine the residuals against the predicted values by seeing how randomly the dots are distributed. There does not appear to be a clear pattern. This supports the assumption of consistency of error across all values of the independent variable.Using the navigation on the left, please proceed to the next page.Multiple Regression: Testing (3 of 3)4. NORMALITY OF THE RESIDUAL. The straight line of the P-P plot is the benchmark. This is how the data would appear if they were “multivariate normal.” Multivariate refers to the fact that we have more than one variable and cannot just look at “the distribution” to see if it is normally distributed. We are looking at the normality of all the predictor variables in combination. So we look at the pattern of dots to see how closely they fit the line. Some of the dots are only slightly off the line at the higher values. Therefore we can say we have met the assumption of normality.5. MULTICOLINEARITY. Refers to the correlation among the IV or predictor variables. Examining the correlation matrix we see that the two independent variables (age and anxiety) have a weak positive correlation, r = .258, p =.054. Concerns for multicolinearity should arise only when correlations are >.70 (+/-). So, we have met this assumption as well.Average number of hours of sleep during weekAgeSelf-rating on Anxiety ScalePearson CorrelationAverage number of hours of sleep during week1.000-.557-.551Age-.5571.000.258Self-rating on Anxiety Scale-.551.0001.000Sig. (1-tailed)Average number of hours of sleep during week.000.000Age.000.054Self-rating on Anxiety Scale.000.054NAverage number of hours of sleep during week404040Age404040Self-rating on Anxiety Scale404040Using the navigation on the left, pl.jpg” alt=”http://myeclassonline.com/ec/courses/AUO_files/AU_img.gif”>Multiple Regression: Effects.png” alt=”http://myeclassonline.com/ec/courses/AUO_files/AU_spacer.gif”>Effects of Variables and SignificanceR2 is the statistic that is used to answer the first question: Do the independent variables predict the dependent variable? We look at this table:ModelRR SquareAdjusted R SquareStd. Error of the EstimateChange StatisticsSig. F ChangeR Square Changedf1df2F Change12-.557(a).699(b).310.488.292.4611.07352.93711.310.17817.09512.868113837.000.001a Predictors: (Constant), Ageb Predictors: (Constant), Age, Self-rating on Anxiety Scalec Dependent Variable: Average number of hours of sleep during weekThere is R, which is the measure of association – in this case, the same as the Pearson r = .-.557. While the Pearson’s r coefficient indicates direction (positive or negative), the same statistic in regression is only looking at strength and not direction. That explains why the R statistic in the Model Summary (b) does not have a negative sign like the Pearson’s r.From the Model Summary we can see that there are two Models, meaning two variables have been entered. The first four columns (in pink) describe how much variance each variable is accounting for. The “Change Statistics” (in blue) indicate if there is a change in impact from one to two variables.Reading across the Model 1 row, the first variable, age, accounts for 31% of the variance (R2 =.310). This is statistically significant, F(1, 38) = 17.095, p <.001.Reading across the Model 2 row, the addition of the second variable, Self-rating on Anxiety Scale, adds just a little more unique variance (R Square Change = .178) so that the total amount of variance explained, R2 using two variables is .488. The amount of variance accounted for is almost 49%. This is also statistically significant, F(1, 37) = 12.868, p = .001.The ANOVA source table tells if the Models are significant.ModelSum of SquaresdfMean SquareFSig.1Regression19.701119.70117.095.000(a)Residual43.793381.152Total63.494392Regression31.001215.50117.651.000(b)Residual32.49237.878Total63.49439a Predictors: (Constant), Ageb Predictors: (Constant), Age, Self-rating on Anxiety Scalec Dependent Variable: Average number of hours of sleep during weekThe F test for the first Model is F(1, 38) = 17.095, p <.001. The model is statistically significant.The F test for the second Model is F(2, 37) = 17.651, p <.001. The model is statistically significant.Regression CoefficientsAnd last, we examine the regression coefficients in the final model. There are two variables, so we can look at ? (Beta) and determine that both predictor variables are significant.Model 1 (in pink): For Age, Beta = -.557 t = -4.135, p <.001.Model 2 (in blue): When the two predictors are added, the Beta of Age changes to -.444, t = -3.650, p = .001. For Anxiety, Beta = -.437, t = -3.587, p = .001.When both predictors are included, both age and anxiousness significantly predict average hours of sleep. Since the value of Beta for Age is slightly higher (we ignore the minus sign), this indicates that Age is slightly more important.Coefficients(a)ModelUnstandardized CoefficientsStandardized Coefficientst.Sig.BStd. ErrorBetaBStd. Error1(Constant)9.915.75813.087.000Age-.068.016-.557-4.135.0002(Constant)10.865.71215.252Age-.054.015-.444-3.650.001Self-rating on Anxiety Scale-.266.074-.437-3.587.001a Dependent Variable: Average number of hours of sleep during weekOur formula for the best fitting line is (B values highlighted in yellow):Y = B1X1 + B2X2 + aY = -.054X1 -.266X2 + 10.865Notice for the formula we use the Unstandardized Coefficients (B).Using the navigation on the left, please proceed to the next page..jpg" alt="http://myeclassonline.com/ec/courses/AUO_files/AU_img.gif">Multiple Regression: Null & Risk Factors.png” alt=”http://myeclassonline.com/ec/courses/AUO_files/AU_spacer.gif”>5. Retain/reject the null hypothesisWe have two research questions and two hypotheses.1. Does the combination of independent variables predict Average Weeknight Sleep?H0: R2 = 0H1: R2>0The first variable, Age, accounts for 31% of the variance. The addition of the second variable, Self-rating on Anxiety Scale, adds a more unique variance (.178) so that the total amount of variance explained, R2 using two variables, is .488, or almost 49% of the variance explained.The model is statistically significant.We reject the null hypothesis.2. Which independent Variables are the best predictors?H0: ? = 0H1: ? ? 0For Model 1 with Age, Beta = .557 t = -4.135, p <.001. We reject the null hypothesis.For Model 2 with Age and Self-Rating of Anxiety: When the two predictors are added, the Beta of Age changes to -.444, t = -3.650, p =.001. For Anxiety, Beta =-.437, t = -3.587, p =.001. Again, we reject the null hypothesis.When both predictors are included, both age and anxiousness significantly predict average hours of sleep. Since the value of Beta for Age is slightly higher (we ignore the minus sign), this indicates that Age is slightly more important than anxiety in predicting sleep. Because the Betas are negative, we know that each has a negative impact on sleep. That is, as age increases, sleep decreases. As anxiety increases sleep decreases.6. Risk of Type I and Type II errorType I ErrorR2 is a measure of effect size, and the interpretation using the effect size chart from previous lessons can be applied here. Notice that it is similar to the discussion of size and strength.Value of dHow strong the effect0.00 to .20No effect to small effect.21 to .33Small to moderate effect.34 to .50Moderately strong effect.51 to .75Strong effect.76 or moreVery strong effectRecall that R2 = .488, or almost 49% of the variance is explained; the remaining 51% is unexplained. R2 is statistically significant, and has a moderately strong effect size.Also, all of the assumptions have been met, suggesting that the risk of Type I error has been minimized.Type II error There are two big risks in multiple regression. One is the risk of unreliable measures which we’ve mentioned before. The other is sample size. In order to have sufficient power, it is recommended that the researcher have 12 to 15 cases per variable in stepwise regression. We have 40 participants, or 20 cases per variable, which is acceptable. So the risk for Type II error is low..jpg" alt="http://myeclassonline.com/ec/courses/AUO_files/AU_img.gif">Multiple Regression: Written Conclusions.png” alt=”http://myeclassonline.com/ec/courses/AUO_files/AU_spacer.gif”>7. State your results in APA style and formatIn writing this up for his dissertation, Mr. Shelby would state:A Multiple Regression analysis was run examining Age and Self-rating on Anxiety Scale as predictors of sleep. Results indicate a statistically significant model, F(2, 37) = 17.651, p <.001. The first variable, Age, accounts for 31% of the variance. The addition of the second variable, Self-rating on Anxiety Scale, contributes 17.8% unique variance; R2 = 49%.Regarding the predictive value of the Independent variables, for Model 1 with Age, Beta = .557 t, = -4.135, p <.001. We reject the null hypothesis. Age is a significant predictor of sleep. When Age and Self-Rating of Anxiety were added, the Beta of Age changes to -.444, t = -3.650, p = .001. For Anxiety, Beta = -.437, t = -3.587, p = .001. Again, we reject the null hypothesis. Thus, amount of sleep is influenced by age and anxiety: the older and more anxious, the less sleep they experience.When both predictors are included, both age and anxiousness significantly predict average hours of sleep. Since the value of Beta for Age is slightly higher (we ignore the minus sign), this indicates that Age is slightly more important.

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