Executive summary

This report summarizes my findings on potential differences between husbands and wives in their ratings of car presentations.  The report includes the following:  1) a brief expla- nation of the study, 2) the data, 4) statistical methods used, and 5) a summary of results.

The study

There is a tendency to associate cars with males, I hypothesize that presentations might inadvertently reflect that kind of bias which could contribute to husbands reacting more favorably to salespersons’ presentations than wives.  To test this, a random sample of 35 couples at the Fast and Furious Ford dealership were asked to give independent ratings. I then used these data to calculate a confidence interval for the mean difference between the husbands’ and wives’ ratings.  if my hypothesis is correct, this confidence interval should lie to the right of zero.

The data

The data collected at the dealership include a husband and a wife for each of the 35 couples in the sample. Exhibit 1 presents data for the first six couples. Summary statistics for the entire data set are given in Exhibit 2. Sample means and medians suggest that husbands tend to rate presentations higher than their wives. However, this comparison doesn’t take into account sample variability.  An inferential statistical method that takes this aspect into account is discussed next.

Statistical methodology

My main objective is comparing two means: the mean rating of husbands and the mean rating of wives. In this course we discussed two statistical methods to compare two means. One of them assumes (among other things) that one has observations from two independent samples. Although husband and wife were asked separately for a presentation assessment, one may suspect that husbands and wives might coincide in their views/opinions, which would result in a positive correlation between their ratings. This suspicion was confirmed by the data.  Figure 1 shows a scatterplot of the data where the x-coordinate represents wife’s rating and y-coordinate gives husband’s rating. The least-squares regression line is also graphed on Figure 1. To supplement this graphical analysis, a measure of the strength of this linear association was computed. The sample correlation between ratings provided by husbands and wives was close to 0.24. Based on this, I decided to use a matched-pairs method instead, which is better suited for data that come in pairs and are positively cor- related.  From what we learned in our course, applying this method will provide us with a narrower confidence interval than the two-sample method. The necessary conditions to apply this method are verified next.

Assumptions

To obtain the desired confidence interval and check that method assumptions are met, R was used (see appendix for more details about R code). First we discuss the assumptions. Above, we have already justified that it is more sensible to consider these two samples as paired data. The sample correlation provides evidence against the independence between

wives’ and husbands’ ratings.   On the other hand, it is reasonable to assume that the assessment of any couple is independent of the assessment of the others, so the differences are independent. Randomization condition is satisfied, our 35 couples are a random sam- ple from the population of interest.  Finally, the assumption of Normality was checked graphically with the aid of a stemplot, Q-Q plot, and Boxplot; see Figures 2, 3, and 4 respectively. All graphs suggest that the Normality assumption is reasonable for the dif- ferences in ratings between husbands and wives.  On one hand, the stemplot shows that the distribution of differences is roughly symmetric and unimodal, with a potential outlier. Boxplot confirms the symmetry of rating differences and doesn’t identify any outliers. The Q-Q plot shows a pattern similar to a step function, which is due to the discrete nature of our observations. However, it doesn’t provide evidence of an important departure from Normality assumption.

Since conditions/assumptions were met, I proceded to construct a T-confidence interval for paired data. Exhibit 3 contains the R code and output. A summary of my conclusions is given below.

Results and Future work

The output provides further evidence that husbands react, on average, more favorably to presentations than their wives. The mean difference is 2.86 points. Graphical representa- tions of the sample differences appear in Figures 2 and 4.  From both of them, it is clear that most of the differences are positive.

The 95% confidence interval for the mean difference (see Exhibit 3) extends from 2.27 to 3.44.  Because the confidence interval doesn’t include zero, we can conclude that the means for husbands and wives are different. Moreover, since the lower confidence limit is positive, this provides further evidence that husbands, on average, react more positively to sales presentations than their wives.

The results mentioned above were obtained under the assumption that a matched-pairs method is more adequate for this data set. The decision of using such a method was based on a point estimate of the correlation coefficient.  An extension of the work presented here, would be carrying out a two-sided hypothesis test for the correlation coefficient and basing the decision on the conclusion of that test. If the null hypothesis that o = 0 can’t be rejected, using a two-sample t procedure  (for independent samples) would be more appropriate.

Appendix

Exhibit 1

Exhibit 2

Exhibit 3

mod=lm(Husband~Wife);

plot(Wife,Husband,xlim=c(0 ,10),ylim=c(0 ,10),

xlab= "Wife's  rating" ,ylab= "Husband's  rating" ,col= "red" ,pch=19);

abline(coef(mod),lty=2 ,col= "blue");

title("Figure  1:  Scatterplot  (Wife's  rating  vs  Husband's  rating)")

Figure 1: Scatterplot (Wife's rating vs Husband's rating)

0                     2                     4                     6                     8                     10

Wife's rating

Figure 2

Figure 3: Q−Q plot for differences

−2                       −1                        0                         1                         2

Theoretical Quantiles