Question

Two critics rate the service at six award-winning restaurants on a continuous 0 to 10 scale. Their rankings are shown in the table below. Restaurant 1 2 3 4 5 6 Critic 1: 6.1 5.2 8.9 7.4 4.3 9.7 Critic 2: 7.3 5.5 9.1 7.0 5.1 9.8

a) Is this paired or unpaired data?

b) Compute a 95% confidence interval for the mean difference in rating. Show all your working.

c) Is there a difference between the critics’ ratings, allowing for some random variation? Answer this question using a parametric hypothesis test, and compare your result to the confidence interval from part b.

d) Use a nonparametric hypothesis test to further investigate whether there is a difference between the critics’ rankings.

e) Is the parametric or non-parametric test more appropriate here?

Answer 1

a) Since the Critic 1 and Critic 2 rate the same restaurant, the given data is paired.

b) Computing the mean difference, d_i:

= -0.367

= 0.561

The 95% confidence interval for the mean difference in rating can be computed as:

= (-0.956, 0.222)

c) To test whether there a difference between the critics’ ratings, allowing for some random variation,

To test: Vs

The appropriate test statistic is given by,

with critical / rejection region

= -1.602

Comparing the test statistic with the critical value for 6 - 1 = 5 df,

Since, |t| = |-1.602| = 1.602 < 2.571 does not lie in the rejection region, we fail to reject the null hypothesis. We may conclude that there is no significant difference between the critics’ ratings.

Also, since, the 95% CI obtained in b), (-0.956, 0.222) includes the null value, , this supports our conclusion that we fail to reject the null hypothesis and that there is no significant difference between the critics’ ratings.

d) The non-parametric equivalent of the parametric paired t test is the Wilcoxon signed rank test:

The test statistic is given by,

with critical region |Z| > 1.96

where SR = Signed Rank

= -0.66

Since, |Z| = 0.66 < 1.96 does not lie in the rejection region, we fail to reject H_0. We may conclude that there is no significant difference between the critics’ ratings.

e) Parametric tests are always preferred over Non parametric test when the data is continuous and is approximately normally distributed. Here, the data is measured in ordinal scale. Hence, although both parametric and non parametric test gives the same result, it would be better if we present the results of a non parametric test for the given ordinal data.