STAT 516 hw 5

Author

Karl Gregory

Students in a class were randomly assigned to receive a piece of paper on which was printed a pie chart, a bar chart, or a stacked bar chart (these are shown below). Students were given five seconds to guess, from looking at the chart, what percentage the chart represented for category “B”.

Here are the guesses:

pie,bar,stacked
15.00,10.00,10.00
20.00,17.00,10.00
14.00,13.00,10.00
15.00,25.00,18.00
15.00,25.00,20.00
14.00,25.00,18.00
15.00,16.67,15.00

It is of interest whether there is any difference in the abilities of these types of charts to effectively convey percentages made up by the parts of a whole. The true percentage associated with the category “B” was $15 %$ . Let $Y_{i j}$ be the absolute deviation from $15 %$ of the guess of student $i$ receiving chart type $j$ , and consider the model that treats these responses as independent random variables normally distributed around their respective means.

1.

Import (however it seems best) the data into R and make side-by-side boxplots of the response values (the responses are not the raw guesses but the absolute deviations of the guesses from $15$ ) for the three chart types.

2.

Give the mean and standard deviation of the response values as well as the number of replications for each chart type.

3.

Compute all quantities in the analysis of variance (ANOVA) table below without using the lm() function.

Source	Df	SS	MS	F value	p-value
Treatment	$a - 1$	${SS}_{Trt}$	${MS}_{Trt}$	$F_{test}$	$P (F > F_{test})$
Error	$N - a$	${SS}_{Error}$	${MS}_{Error}$
Total	$N - 1$	${SS}_{Tot}$

4.

Obtain the ANOVA table using the lm() and anova() function (from here on you may use the lm() function).

5.

Give the null hypothesis for which the value $F_{test}$ in the ANOVA table is the test statistic.

6.

State your conclusions with respect to the null hypothesis in the previous part.

7.

Check whether the assumptions of the one-way ANOVA model are satisfied.

8.

Conduct Levene’s test for equal variances in the response values across chart types.

9.

Give the values of $\hat{μ}$ and of the ${\hat{τ}}_{i}$ in the estimated model and interpret their values.

10.

Proceeding as though the assumptions were satisfied, perform a careful analysis in order to determine, if possible, which of the three chart types is best for giving an accurate impression of a percentage. Give a careful summary of your findings.

11.

Now take the pie chart as a benchmark against which the other two chart types are to be compared. Investigate carefully whether the ability of the pie chart to convey a percentage is different from that of the bar and the stacked bar charts. Carefully summarize your findings.

12.

Should we have expected the findings from parts 10. and 11. to be very different? Why or why not?