Stat 3300 Homework 9 Due Thursday 05282020 Notes And Answers

Stat 3300 Homework 9due Thursday 05282020note Answer These Quest

Answer these questions on a separate piece of paper. In the top right corner, include your name, SMU ID, and course number. Please include a title for the assignment so it is clear to the graders. If you miss class the day the assignment is turned in, submit this before class in order to receive credit.

Question 1 (35 points total, 7 points each):

According to William Butler Yeats, "She is the Gaelic muse, for she gives inspiration to those she persecutes. The Gaelic poets die young, for she is restless, and will not let them remain long on earth." One study designed to investigate this issue examined the age at death for writers from different cultures and genders. Three categories of writers examined were novelists, poets, and nonfiction writers. Most of the writers are from the United States, but Canadian and Mexican writers are also included. The data are given in "ex12-46poets.csv". Use R to answer the following:

• a) Examine the assumptions necessary for ANOVA. Summarize your findings.
• b) Run the ANOVA and report the results. Include the null and alternative hypotheses, the test statistic, the p-value, and a conclusion in the context of the problem.
• c) Use a contrast to compare the poets with the two other types of writers. Use a one-sided alternative based on the quote from Yeats (poets die at a younger age). Note that the levels of "Type" are non-fiction, novels, and poems; non-fiction is group 1, novels is group 2, and poems is group 3.
• d) Use another contrast to compare the novelists with the non-fiction writers.
• e) Construct a boxplot for the three groups of writers. Use the Bonferroni procedure to compare the three means (α = 0.05).

The data are in "ex12-46poets.csv".

Question 2 (32 points total, 8 points each):

Each of the following tables gives means for a two-way ANOVA. Make a plot of the means with the levels of Factor A on the x-axis. State whether or not there is an interaction, and if there is, describe it.

Question 3 (33 points total):

In a study of bilingualism and aging, 80 participants were divided into four groups: younger adult bilinguals, older adult bilinguals, younger adult monolinguals, and older adult monolinguals. Participants completed color-shape task-switching tests, with total reaction time recorded in microseconds; shorter times indicate better performance. The data are in "EX13-22BILING.csv".

• a) (9 points) Make a table giving the sample size, mean, and standard deviation for each group. Is it reasonable to pool the variances?
• b) (9 points) If bilingualism helps with brain functioning as we age, explain why we’d expect to find an interaction between age and lingualism. Also, create an interaction plot illustrating the expected pattern.
• c) (15 points) Analyze the data using an ANOVA. Report the test statistics, degrees of freedom, and p-values (α = 0.10). Summarize your findings in a short paragraph.

Paper For Above instruction

The following analysis explores multiple research questions using ANOVA across different datasets, examining the effects of categorical variables on response variables, assumptions underlying these tests, and their interpretations in context.

Question 1: The study investigates whether there is a significant difference in age at death among writers classified as non-fiction, novels, and poets. Initial assumptions for ANOVA include normality of residuals, homogeneity of variances, and independence of observations. To examine normality, a Shapiro-Wilk test or histogram analysis of residuals can be employed; homogeneity of variances can be assessed via Levene's test, and independence is assured through the study design. Summaries typically indicate whether these assumptions are reasonably met or if data transformations are necessary.

Running the ANOVA involves setting hypotheses: H0 (no difference among group means) and Ha (at least one group mean differs). The F-test statistic, degrees of freedom, and p-value are obtained; if the p-value is less than 0.05, we reject H0, concluding significant differences exist. In the context of the study, a significant ANOVA suggests that genre influences age at death, aligning with Yeats’s poetic reflection about poets dying young.

Contrasts are used to explore specific hypotheses. A contrast comparing poets to other writers tests if poets die younger overall, employing coefficients to weight group means accordingly, with a one-sided alternative reflecting the expectation that poets have lower mean ages. Similarly, comparing novelists with non-fiction writers illuminates differences between these groups. The construction of contrasts involves defining contrast coefficients and calculating contrast estimates and standard errors to assess their significance.

The boxplot visually displays differences across the three groups, revealing median differences, spread, and potential outliers. Applying the Bonferroni correction adjusts for multiple comparisons, maintaining overall α at 0.05 while performing pairwise mean comparisons to determine which groups differ significantly.

Question 2: The table of means in a two-way ANOVA examines whether the interaction between factors is significant. Plotting group means against levels of Factor A highlights potential interaction patterns. An interaction effect suggests that the effect of one factor depends on the level of another, which can be described in terms of crossing lines or non-parallel slopes in the interaction plot. This indicates that differences among levels are not consistent across another factor, providing insight into combined effects.

Question 3: The analysis assesses the effect of age, language, and their interaction on reaction times. Sample sizes, means, and standard deviations inform whether variances are similar enough to justify pooling via Levene’s test or similar procedures. An interaction between age and language would imply that the benefit of bilingualism in reaction times varies with age, expected to manifest as a non-parallel interaction plot pattern: perhaps younger bilinguals perform best, older monolinguals perform worst, with other groups demonstrating intermediate trends.

The ANOVA results include F-statistics, degrees of freedom, and p-values, interpreted to understand main effects and interactions. Significant interactions indicate that age and language status do not influence reaction time independently but interactively, influencing the observed outcomes, consistent with hypotheses about cognitive reserve and neuroplasticity in aging.

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