Posted: February 24th, 2022

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Single Factor ANOVA is a method we use when we want to compare a quantitative variable between more than two categories. The procedure evaluates whether the means of different treatment groups, or populations, are different for at least one of the populations. When we only have two populations, we can perform a two-sample t-procedure; however, when we have more than two populations, we need to examine the data with the Single Factor ANOVA procedure.

California, New York, and Texas are three of the most populated states in the United States, and their four-year universities attract students from across the world. A researcher is interested in comparing the average in-state tuition for four-year universities in California, New York, Texas, and Oregon (because we are enrolled at Oregon State University), so they took a random sample of 35 four-year universities from each of the four states and recorded their annual in-state tuition cost in 2020-2021 (in USD$).

The R script DA6_SingleFactor_ANOVA.R provides an analysis that compares average annual cost of out-of-state tuition in the 2020-2021 school year across four states: California (abbreviated CA), New York (abbreviated NY), Texas (abbreviated TX), and Oregon (abbreviated OR). You will need to download and then upload the dataset, tuition.csv, in R. Note that you do need to edit the R code in order to perform an analysis on the in-state tuition. Units are in US dollars. Use a significance level of =0.10

a. (3 points) Create side-by-side boxplot of the data, adding color and appropriate titles/labels to your plot. Paste your plot here. Make sure youve edi2020-2021 ted the code so that it is IN-STATE.

b. (2 points) Looking at the side-by-side boxplot, does there appear to be a difference between the mean tuition between the four states? Explain your reasoning.

c. (3 points) State the appropriate null and alternative hypotheses for the Single Factor ANOVA F test. Make sure youve edited the code so that it is IN-STATE.

d. (3 points) State each of the conditions for the Single Factor ANOVA F-Test, as well as whether each condition is reasonably satisfied. If the conditions are not met, still proceed with the analysis.

e. Perform the Single Factor ANOVA F-test in R. Make sure youve edited the code so that it is for IN-STATE.

i. (2 point) Paste the ANOVA table here (make sure that it is easy to read).

ii. (2 points) From the ANOVA table, what is the average between treatment/group variability (MSTr)?

iii. (2 points) From the ANOVA table, what is the average within group variability (MSE)?

f. (6 points) Use the F-statistic and p-value from the ANOVA table to state whether there is a significant difference between at least two of the group means. Here, we want the two-part conclusion for the hypothesis test. Make sure youve edited the code so that it is IN-STATE.

Your conclusion should include:

the test statistic and its degrees of freedom

the p-value

whether you reject or fail to reject the null hypothesis

a statement in terms of the strength of evidence in terms of the alternative, including context.

g. (2 points) Suppose the researcher found that the average California in-state tuition was statistically significantly different from one of the other states. Can they conclude from this analysis that it is the state that universities are located within that caused the difference in mean tuition that they observed? Explain why or why not, based on what we know about the design of this study. (Hint: reading through Week 6 material on comparative study design might come in handy here)

h. Now we want to look at all of the pairwise comparisons for the four states. Use the Tukeys Multiple Comparison procedure output in R. We are interested in whether there are any individual comparisons that are significant at the 0.10 significance level. Make sure youve edited the code so that it is IN-STATE.

i. (2 points) Paste R output, the table and the plot, for the multiple comparisons procedure (make sure that it is easy to read). Make sure youve edited the code so that it is IN-STATE.

ii. (2 points) List all comparisons that are significant (or state the comparisons that are not significantmake it very clear which you are listing) at the 0.10 significance level. Make sure youve edited the code so that it is IN-STATE.

iii. (6 points) Looking at the 90% familywise confidence intervals that R calculated, take the interval associated with the smallest p-value and interpret that particular 90% familywise confidence interval for the difference in context of the problem (think two-part conclusion for a confidence interval). Make sure youve edited the code so that it is IN-STATE

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