One way ANOVA, statistics homework help

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Mathematics

Description

For this assessment, you will complete an SPSS data analysis report using a one-way ANOVA for assigned variables.

We are often confronted with a need to compare the means of more than two groups. We may also need to compare more than two scores among a sample of participants. ANOVA, ANalysisOf VAariance, is a technique designed to address such questions.

By successfully completing this assessment, you will demonstrate your proficiency in the following course competencies and assessment criteria:

  • Competency 1: Analyze the computation, application, strengths and limitations of various statistical tests.
    • Develop a conclusion, including strengths and limitations of a one-way ANOVA.
  • Competency 2: Analyze the decision making process of data analysis.
    • Analyze the assumptions of a one-way ANOVA.
  • Competency 3: Apply knowledge of hypothesis testing.
    • Articulate a research question, null hypothesis, alternative hypothesis, and alpha level.
  • Competency 4: Interpret the results of statistical analyses.
    • Interpret the one-way ANOVA output.
  • Competency 5: Apply a statistical program's procedure to a data set.
    • Apply the appropriate SPSS procedures to check assumptions and calculate the one-way ANOVA to generate relevant output.
  • Competency 6: Apply the results of statistical analyses to a field of interest or career.
    • Develop a context for the data set, including a definition of required variables and scales of measurement.
  • Competency 7: Communicate in a manner that is scholarly, professional, and consistent with the expectations for members in the identified field of study.
    • Communicate in a manner that is scholarly, professional, and consistent with expectations for members of the identified field of study.

Read the Assessment 4 Context document for important information on the following topics:

  • The logic of a one-way ANOVA.
  • Avoiding inflated Type I error.
  • Hypothesis testing in one-way ANOVA.
  • Assumptions of one-way ANOVA.
  • As you prepare to complete this assessment, you may want to think about other related issues to deepen your understanding or broaden your viewpoint. You are encouraged to consider the questions below and discuss them with a fellow learner, a work associate, an interested friend, or a member of your professional community. Note that these questions are for your own development and exploration and do not need to be completed or submitted as part of your assessment.

APPLICATION OF F-TESTS

  • Is there a research question from your professional life or career specialization that can be addressed by a one-way ANOVA?
  • Why would a one-way ANOVA be the appropriate analysis for this research question?
  • What is the expected outcome?

REQUIRED RESOURCES

The following resources are required to complete the assessment.

In addition, you will need the grades.savfile that you created from the grades2.datfile in Assessment 1.

SPSS Software

Capella University requires learners to meet certain minimum computer requirements. Please note that some software required for a course may exceed these minimum requirements. The following statistical analysis software is required to complete your assessments in this course:

  • IBM SPSS Statistics (recent version for PC or Mac).

This course requires the following as a minimum:

  • IBM SPSS Statistics Standard GradPack. (The Base GradPack is not acceptable for use in this course.)
Internet Resources

PREPARATION

Read the Assessment 4 Context document (linked in the Resources, under the Required Resources heading) to learn about the concepts used in this assessment.

You will use the following resources for this assessment. They are linked in the Required Resources.

  • Complete this assessment using the DAA Template.
  • Read the SPSS Data Analysis Report Guidelines for a more complete understanding of the DAA Template and how to format and organize your assessment.
  • Refer to IBM SPSS Step-By-Step Instructions: One-Way ANOVA for additional information on using SPSS for this assessment.
  • If necessary, review the Copy/Export Output Instructions to refresh your memory on how to perform these tasks. As with your previous assessments, your submission should be narrative with supporting statistical output (table and graphs) integrated into the narrative in the appropriate place (not all at the end of the document).

You will analyze the following variables in the grades.sav data set:

  • section.
  • quiz3.

DIRECTIONS

Step 1: Write Section 1 of the DAA
  • Provide a context of the grades.sav data set.
  • Include a definition of the specified variables (predictor, outcome) and corresponding scales of measurement.
  • Specify the sample size of the data set.
Step 2: Write Section 2 of the DAA
  • Analyze the assumptions of the one-way ANOVA.
  • Paste the SPSS histogram output for quiz3 and discuss your visual interpretations.
  • Paste SPSS descriptives output showing skewness and kurtosis values for quiz3 and interpret them.
  • Paste SPSS output for the Shapiro-Wilk test of quiz3 and interpret it.
  • Report the results of the Levene's test and interpret it.
  • Summarize whether or not the assumptions of the one-way ANOVA are met.
Step 3: Write Section 3 of the DAA
  • Specify a research question related to the one-way ANOVA.
  • Articulate the null hypothesis and alternative hypothesis.
  • Specify the alpha level.
Step 4: Write Section 4 of the DAA
  • Begin by pasting SPSS output of the means plot and providing an interpretation.
  • Also report the means and standard deviations of quiz3 for each level of section.
  • Next, paste the SPSS ANOVA output and report the results of the F test, including:
    • Degrees of freedom.
    • Fvalue.
    • pvalue.
    • Calculated effect size.
    • Interpretation of the effect size.
  • Finally, if the omnibus F is significant, provide the SPSS post-hoc (Tukey HSD) output.
    • Interpret the post-hoc tests.
Step 5: Write Section 5 of the DAA
  • Discuss the conclusions of the one-way ANOVA as it relates to the research question.
  • Conclude with an analysis of the strengths and limitations of one-way ANOVA.

Unformatted Attachment Preview

IBM SPSS Step-by-Step Guide: One-Way ANOVA Note: This guide is an example of creating ANOVA output in SPSS with the grades.sav file. The variables shown in this guide do not correspond with the actual variables assigned in Assessment 4. Carefully follow the instructions in the assignment for a list of assigned variables. Screen shots were created with SPSS 21.0. Creating One-Way ANOVA Output To complete Section 2 of the DAA for Assessment 4, you will generate SPSS output for a histogram, descriptive statistics, and the Shapiro-Wilk test, which are covered in previous step-by-step guides. The Levene test (homogeneity of variance) is covered in the steps below. Refer to the Assessment 4 instructions for a list of assigned variables. The example variables year and final are shown below. Step 1. Open grades.sav in SPSS. 1 Step 2. On the Analyze menu, point to Compare Means and click One-Way ANOVA… Step 3. In the One-Way ANOVA dialog box: • Move the assigned dependent variable into the Dependent List box. • Move the assigned independent variable into the Factor box. The examples of final and year are shown below. • Click the Options button. 2 Step 4. In the One-Way ANOVA: Options dialog box: • Select Homogeneity of variance test (for the Levene test for Section 2 of the DAA). • Select Descriptive and Means Plot (for Section 4 of the DAA). • Click Continue. • Return to the One-Way ANOVA dialog box and select the Post Hoc button. Step 5. In the One-Way ANOVA: Post Hoc Multiple Comparisons dialog box: • Check the Tukey option for multiple comparisons. • Click Continue and OK. 3 Interpreting One-Way ANOVA Output A string of ANOVA output will appear in SPSS. (The output below is for the example variable final.) Step 1. Copy the Levene test output from SPSS and paste it into Section 2 of the DAA Template. Then interpret it for the homogeneity of variance assumption. Test of Homogeneity of Variances final Levene Statistic .866 df1 df2 3 Sig. 101 .462 Step 2. Copy the means plot, paste it into Section 4 of the DAA Template, and interpret it. Step 3. Copy the descriptives output. Paste it into Section 4 along with the report of means and standard deviations of the dependent variable at each level of the independent variable. Descriptives final N Mean Std. Deviation Std. Error 95% Confidence Interval for Minimum Maximum Mean Lower Bound Upper Bound Frosh 3 59.33 5.859 3.383 44.78 73.89 55 66 Soph 19 62.42 6.628 1.520 59.23 65.62 48 72 Junior 64 61.47 8.478 1.060 59.35 63.59 40 75 Senior 19 60.89 7.951 1.824 57.06 64.73 43 74 105 61.48 7.943 .775 59.94 63.01 40 75 Total 4 Step 4. Copy the ANOVA output, paste it into Section 4, and interpret it. ANOVA final Sum of Squares Between Groups df Mean Square F 37.165 3 12.388 Within Groups 6525.025 101 64.604 Total 6562.190 104 Sig. .192 .902 Step 5. Finally, if the overall ANOVA is significant, copy the post hoc output, paste it into Section 4, and interpret it. Multiple Comparisons Dependent Variable: final Tukey HSD (I) Year in school (J) Year in school Mean Std. Error Sig. Difference (I-J) Frosh Soph Junior Senior 95% Confidence Interval Lower Bound Upper Bound Soph -3.088 4.993 .926 -16.13 9.96 Junior -2.135 4.748 .970 -14.54 10.27 Senior -1.561 4.993 .989 -14.61 11.48 Frosh 3.088 4.993 .926 -9.96 16.13 Junior .952 2.100 .969 -4.53 6.44 Senior 1.526 2.608 .936 -5.29 8.34 Frosh 2.135 4.748 .970 -10.27 14.54 Soph -.952 2.100 .969 -6.44 4.53 Senior .574 2.100 .993 -4.91 6.06 Frosh 1.561 4.993 .989 -11.48 14.61 Soph -1.526 2.608 .936 -8.34 5.29 Junior -.574 2.100 .993 -6.06 4.91 5 SPSS Data Analysis Report Guidelines For the SPSS data analysis report assignments in Assessments 2, 3, and 4, you will use the Data Analysis and Application (DAA) Template with the five sections described below. As shown in the IBM SPSS step-by-step guides, label all tables and graphs in a manner consistent with APA Style and Format guidelines. Citations, if needed, should be included in the text and references included in a reference section at the end of the report. The organization of the report should include the following five sections: Section 1: Data File Description (One Paragraph) 1. Describe the context of the data set. Cite a previous description if the same data set is used from a previous assignment. To increase the formal tone of the DAA, avoid firstperson perspective "I." For example, do not write, "I ran a scatter plot shown in Figure 1." Instead, write, "Figure 1 shows. . . ." 2. Specify the variables used in this DAA and the scale of measurement of each variable. 3. Specify sample size (N). Section 2: Testing Assumptions (Multiple Paragraphs) 1. Articulate the assumptions of the statistical test. 2. Paste SPSS output that tests those assumptions and interpret them. Properly embed SPSS output where appropriate. Do not string all output together at the beginning of the section. In other words, interpretations of figures and tables should be near (that is, immediately above or below) where the output appears. Format figures and tables per APA formatting. Refer to the examples in the IBM SPSS step-by-step guides. 3. Summarize whether or not the assumptions are met. If assumptions are not met, discuss how to ameliorate violations of the assumptions. Section 3: Research Question, Hypotheses, and Alpha Level (One Paragraph) 1. Articulate a research question relevant to the statistical test. 2. Articulate the null hypothesis and alternative hypothesis for the research question. 3. Specify the alpha level (.05 unless otherwise specified). Section 4: Interpretation (Multiple Paragraphs) 1. Paste SPSS output for an inferential statistic and report it. Properly embed SPSS output where appropriate. Do not string all output together at the beginning of the section. In other words, interpretations of figures and tables should be near (that is, immediately above or below) where the output appears. Format figures and tables per APA formatting. 2. Report the test statistics. For guidance, refer to the "Results" examples at the end of the appropriate chapter of your Warner text. 3. Interpret statistical results against the null hypothesis. Section 5: Conclusion (Two Paragraphs) 1. Provide a brief summary (one paragraph) of the DAA conclusions. 2. Analyze strengths and limitations of the statistical test. Running head: DATA ANALYSIS AND APPLICATION TEMPLATE Data Analysis and Application (DAA) Template Learner Name University 1 DATA ANALYSIS AND APPLICATION TEMPLATE Data Analysis and Application (DAA) Template Use this file for all assignments that require the DAA Template. Although the statistical tests will change from week to week, the basic organization and structure of the DAA remains the same. Update the title of the template. Remove this text and provide a brief introduction. Section 1: Data File Description 1. Describe the context of the data set. You may cite your previous description if the same data set is used from a previous assignment. 2. Specify the variables used in this DAA and the scale of measurement of each variable. 3. Specify sample size (N). Section 2: Testing Assumptions 1. Articulate the assumptions of the statistical test. 2. Paste SPSS output that tests those assumptions and interpret them. Properly integrate SPSS output where appropriate. Do not string all output together at the beginning of the section. 3. Summarize whether or not the assumptions are met. If assumptions are not met, discuss how to ameliorate violations of the assumptions. Section 3: Research Question, Hypotheses, and Alpha Level 1. Articulate a research question relevant to the statistical test. 2. Articulate the null hypothesis and alternative hypothesis. 3. Specify the alpha level. Section 4: Interpretation 1. Paste SPSS output for an inferential statistic. Properly integrate SPSS output where appropriate. Do not string all output together at the beginning of the section. 2. Report the test statistics. 2 DATA ANALYSIS AND APPLICATION TEMPLATE 3. Interpret statistical results against the null hypothesis. Section 5: Conclusion 1. State your conclusions. 2. Analyze strengths and limitations of the statistical test. 3 DATA ANALYSIS AND APPLICATION TEMPLATE References Provide references if necessary. 4 Print Copy/Export Output Instructions SPSS output can be selectively copied and pasted into Word by using the Copy command: 1. 2. 3. 4. Click on the SPSS output in the Viewer window. Right-click for options. Click the Copy command. Paste the output into a Microsoft Word document. The Copy command will preserve the formatting of the SPSS tables and charts when pasting into Microsoft Word. An alternative method is to use the Export command: 1. 2. 3. 4. 5. Click on the SPSS output in the Viewer window. Right-click for options. Click the Export command. Save the file as Word/RTF (.doc) to your computer. Open the .doc file. Assessment 4 Context Recall that null hypothesis tests are of two types: (1) differences between group means and (2) association between variables. In both cases there is a null hypothesis and an alternative hypothesis. In the group means test, the null hypothesis is that the two groups have equal means, and the alternative hypothesis is that the two groups do not have equal means. In the association between variables type of test, the null hypothesis is that the correlation coefficient between the two variables is zero, and the alternative hypothesis is that the correlation coefficient is not zero. Notice in each case that the hypotheses are mutually exclusive. If the null is false, the alternative must be true. The purpose of null hypothesis statistical tests is generally to show that the null has a low probability of being true (the p value is less than .05) – low enough that the researcher can legitimately claim it is false. The reason this is done is to support the allegation that the alternative hypothesis is true. In this context you will be studying the details of the first type of test again, with the added capability of comparing the means among more than two group at a time. This is the same type of test of difference between group means. In variations on this model, the groups can actually be the same people under different conditions. The main idea is that several group mean values are being compared. The groups each have an average score or mean on some variable. The null hypothesis is that the difference between all the group means is zero. The alternative hypothesis is that the difference between the means is not zero. Notice that if the null is false, the alternative must be true. It is first instructive to consider some of the details of groups. One might ask why we would not use multiple t tests in this situation. For instance, with three groups, why would I not compare groups one and two with a t test, then compare groups one and three, and then compare groups two and three? The answer can be found in our basic probability review. We are concerned with the probability of a TYPE I error (rejecting a true null hypothesis). We generally set an alpha level of .05, which is the probability of making a TYPE I error. Now consider what happens when we do three t tests. There is .05 probability of making a TYPE I error on the first test, .05 probability of the same error on the second test, and .05 probability on the third test. What happens is that these errors are essentially additive, in that the chances of at least one TYPE I error among the three tests much greater than .05. It is like the increased probability of drawing an ace from a deck of cards when we can make multiple draws. ANOVA allows us do an "overall" test of multiple groups to determine if there are any differences among groups within the set. Notice that ANOVA does not tell us which groups among the three groups are different from each other. The primary test in ANOVA is only to determine if there is a significant difference among the groups somewhere. You will study the theory and logic of analysis of variance (ANOVA). Recall that a t-test requires a predictor variable that is dichotomous. The advantage of ANOVA over a t-test is that the categorical predictor variable includes 3+ values (groups). Just like a t-test, the outcome variable in ANOVA is quantitative and requires the calculation of group means. In ANOVA, there are two levels of hypotheses. There is first the overall question of whether all the group means are equal, or if there are some differences among the means somewhere. This is called the omnibus null hypothesis test. The test is designed to show that the probability that the group means are all equal is very low, leading to the researcher being able to legitimately claim there are differences. This is done with the F test. In ANOVA, once the omnibus null 1 hypothesis is rejected, then one may legitimately use special tests, called post hoc tests, to examine each of the pairs of groups in the set to determine which ones differ and which do not. For instance, if an ANOVA is performed for three groups, the omnibus null hypothesis is that the three groups have equal means. If that null is rejected, then the researcher may use special post hoc tests to compare groups 1 & 2, groups 1 & 3, and groups 2 & 3. Each of these post hoc tests are themselves null hypothesis tests, similar to the t tests which were studied previously. They are designed to control for multiple comparisons, or an inflation of the Type I error rate that is a result of doing many tests with some fixed probability of error on each test. Most are based on the assumption that the omnibus null has been rejected. The Logic of a One-Way ANOVA The ANOVA, or F-test, relies on predictor variables referred to as factors. A factor is a categorical (nominal) predictor variable. The term "one-way" is applied to an ANOVA with only one factor that is defined by two or more mutually exclusive groups. Technically, an ANOVA can be calculated with only two groups, but the t-test is usually used instead. The one-way ANOVA is usually calculated with three or more groups, which are often referred to as levels of the factor. If the ANOVA includes multiple factors, it is referred to as a factorial ANOVA. An ANOVA with two factors is referred to as a "two-way" ANOVA; an ANOVA with three factors is referred to as a "three-way" ANOVA, and so on. Factorial ANOVA is studied in Advanced Inferential Statistics. In this course, we will focus on the theory and logic of the one-way ANOVA. ANOVA is one of the most popular statistics used in psychological research. In nonexperimental designs, the one-way ANOVA compares group means across naturally existing characteristics of groups, such as political affiliation. In experimental designs, the one-way ANOVA compares group means for participants randomly assigned to treatment conditions (for example, high caffeine dose; low caffeine dose; control group). Avoiding Inflated Type I Error You may wonder why a one-way ANOVA is necessary. For example, if a factor has four groups ( k = 4), why not just run independent sample t tests for all pairwise comparisons (for example, Group A versus Group B, Group A versus Group C, Group B versus Group C, et cetera)? Warner (2013) points out that a factor with four groups involves six pairwise comparisons. The issue is that conducting multiple pairwise comparisons with the same data leads to inflated risk of a Type I error (incorrectly rejecting a true null hypothesis—getting a false positive). The ANOVA protects the researcher from inflated Type I error by calculating a single omnibus test that assumes all k population means are equal. Although the advantage of the omnibus test is that it helps protect researchers from inflated Type I error, the limitation is that a significant omnibus test does not specify exactly which group means differ, just that there is a difference "somewhere" among the group means. A researcher therefore relies on either (a) planned contrasts of specific pair wise comparisons determined prior to running the F-test, or, (b) follow-up tests of pair wise comparisons, also referred to as post-hoc tests, to determine exactly which pair wise comparisons are significant. Usually, if planned contrasts are designed correctly, there is no need to perform the omnibus null test, and the overall ANOVA is not necessary. 2 Hypothesis Testing in One-Way ANOVA The null hypothesis of the omnibus test is that all k population means are equal, or H0: µ1 = µ2 =…µk. By contrast, the alternative hypothesis is usually articulated by stipulating that H0 is not true. Keep in mind that this prediction does not imply that all groups must significantly differ from one another on the outcome variable. In fact, for reasons beyond the scope of our present discussion, it is not even strictly necessary that any two groups differ even if the omnibus null is rejected. Assumptions of One-Way ANOVA The assumptions of ANOVA reflect assumptions of the t-test. ANOVA assumes independence of observations. ANOVA assumes that outcome variable Y is normally distributed. ANOVA assumes that the variance of Y scores is equal across all levels (groups) of the factor. These ANOVA assumptions are checked in the same process used to check assumptions for the t-test discussed earlier in the course—using the Shapiro-Wilk test and the Levene test. Effect Size for a One-Way ANOVA The effect size for a one-way ANOVA is eta squared (η2). It represents the amount of variance in Y that is attributable to group differences. Recall the concept of sum of squares ( SS). Eta squared for the one-way ANOVA is calculated by dividing the sum of squares of between-group differences (SS-between) by the total sums of squares in the model (SS-total), which is reported in SPSS output for the F-test. Eta squared for the one-way ANOVA is interpreted with .06 as "large." References Lane, D. M. (2013). HyperStat online statistics textbook. Retrieved from http://davidmlane.com/hyperstat/index.html Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand Oaks, CA: Sage Publications. 3
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Running Head: ONE WAY ANOVA

1

One Way ANOVA:
Comparing Quiz score across three sections
Name:
University
April 16, 2017

2
Section 1
The dataset grade.sav provides information about the demographics of the students
and their final, quiz, gpa and total score for three different sections. The aim of the paper is to
compare the quiz score among the three sections using one-way ANOVA. Here, the predictor
or the independent variable is the student’s section which is qualitative and categorical and
measured on a nominal scale. The outcome variable is the quiz score which is the dependent
variable in this context.The sample size used for the analysis consisted of 105 students where
33 belonged to section 1,39 belonged to section 2 and 33 belonged to section 3.
Section 2
Independence of observations
It is of utmost important to analyze the assumptions of ANOVA before comparing the
mean quiz score among three different sections. The first assumptions require that the
observations are independent. It is also correct in the present scenario as the quiz score of
each is not affected by other’s student quiz score.
Normal distribution of Outcome variable
The ANOVA also require that the outcome variable is normally distributed. The
normal distribution is analysed through histogram plot, skewness value and finally through
Shapiro-Wilk test.
Histogram Plot: The histogram plot is shown in Fig 1 below. The plot shows that the
quiz score is skewed to the left and thus the assumption of normal distribution is not valid. To
analyze further the skewness value is measured and compared with it’s standard error of
skewness value. (Ghasemi and Zahediasl,2017).

3

Fig 1.Histogram plot of quiz 3

Descriptive Statistics: The descriptive statistics was computed using SPSS the result
of which is shown in Table 1 below.(George and Mallery,2007). The mean quiz score
obtained was 7.98 with the average deviation from the mean being 2.308.The skewness is 1.134 indicating quiz score being skewed to the left. The kurtosis is 0.750 having longer and
flatter distribution compared to a normal distribution. It can be seen from Table 1 that the
absolute value of skewness(1.134) is greater than twice its standard error and hence
approximate normal assumption does not hold good in this context. Thus the normal
assumption is not valid. Finally, Shapiro-Wilk test was used to test the deviation of quiz score
from normal distribution.
Table 1
Descriptive Statistics
N

Mean

Std. Deviation

Statistic

Statistic

Statistic

quiz3

105

Valid N (listwise)

105

7.98

2.308

Skewness
Statistic
-1.134

Kurtosis

Std. Error
.236

Statistic
.750

Std. Error
.467

4
Shapiro-Wilk Test: The output of the Shapiro-Wilk test of normality is shown in
Table 2.The null hypothesis is that the quiz score is normally distributed. Assuming a
significance level of 0.05, the p-value obtained for the test was less than 0.05 and conclude
that the quiz score is significantly different from normal.
Table 2.
Tests of Normality
Kolmogorov-Smirnova
Statistic
quiz3

df

.213

Shapiro-Wilk

Sig.
105

Statistic

.00...


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