Question

An experiment is conducted to compare two new automobile designs. Twenty people are randomly selected, and each person is asked to rate each design on a scale of 1 (poor) to 10 (excellent). The resulting ratings will be used to test the null hypothesis that the mean level of approval is the same for both designs against the alternative hypothesis that one of the automobile designs is preferred. Do these data satisfy the assumptions required for the Student's \(t\) -test of Section 10.4 ? Explain.

Short answer

Based on the given information, evaluate whether it is appropriate to use the Student's t-test for this experiment in comparing the two automobile designs' ratings. Additionally, discuss the importance of checking the assumptions for a statistical test before conducting it.

Step-by-step solution

Assess Data Independence

In the experiment, 20 people are asked to rate each automobile design independently. Since each rating is provided by a different individual, we can assume that the data points are independent within the same automobile design. However, since the same individuals are rating both automobile designs, the ratings for the two designs may be correlated. For example, if a person gives high ratings to one of the designs, they may also give high ratings to the other design, or vice versa. Due to this possible dependence, the assumption of data independence may not be fully satisfied.

Check for Normal Distribution

The exercise does not provide specific data, so we cannot perform statistical tests (e.g., Shapiro-Wilk or Anderson-Darling) to check if the data follows a normal distribution specifically. However, since we are dealing with a continuous variable (ranging from 1 to 10) and a relatively reasonable sample size (20 individuals rating each design), it is plausible to assume that the data may approximate a normal distribution. If the exercise provided actual data, performing a normality test would be essential to confirm this assumption.

Test for Equal Variances

Similarly to the normal distribution assumption, the exercise does not provide specific data, so we cannot perform a statistical test (e.g., Levene's test or Bartlett's test) to check if the variances of the two populations being compared (ratings for each automobile design) are equal. Without the actual data, we cannot confirm whether this assumption is met or not. To summarize, we cannot confirm whether the data satisfies all assumptions required for the Student's t-test based on the given information. However, we have reasons to believe that the data may not be completely independent due to the same individuals rating both designs. The normal distribution and equal variances assumptions are plausible but cannot be tested without access to actual data.

Key concepts

Data Independence

For a t-test to be valid, one essential condition is data independence. This means that the ratings or measurements collected must not influence each other. In the context of the given exercise, each individual's rating should only reflect their personal opinion, without any bias from other participants. This would ensure that the data points are independent.

Independence is a crucial factor because any dependence between observations can skew the results of our statistical test, leading to inaccurate conclusions. In this exercise, although the ratings for each car design are provided by unique individuals, all individuals rate both designs. This setup inherently introduces potential correlation. For example, if a person tends to give high ratings, they might do so for both designs, which means their ratings are not independent.

To better satisfy this assumption in a practical scenario, researchers could ensure that individuals rate different car models separately or even use different groups of participants for each design.

Normal Distribution Assumption

The assumption of normal distribution is central to the t-test because it relies on the data being approximately normally distributed. This assumption helps ensure that the test results are statistically valid and applicable to the population.

With a sample size of 20 participants grading each design, the assumption of an underlying normal distribution is reasonable, though it cannot be verified without actual data. Such a sample size is often considered enough for the Central Limit Theorem to apply, which suggests that, for large enough samples, the sampling distribution of the mean will be normal regardless of the distribution of the population.

If actual data were available, tests like the Shapiro-Wilk test could verify normality. Additionally, graphical techniques, such as Q-Q plots, can visually assess the normality of data. Regardless, it is always important to approach assumptions of normality with caution, especially with smaller sample sizes.

Equal Variances Assumption

Another critical assumption for the t-test is that the variances in each group being compared are equal. In statistical terminology, this is known as homogeneity of variances. This ensures that the variability within each group is similar enough for the t-test to compare them fairly.

Without specific data, we cannot directly test this assumption. However, in situations where we have access to actual numbers, tests like Levene's test or Bartlett's test could be used to assess whether variance equality holds.

Why is this important? If the variances are not equal, it can lead to incorrect conclusions about the statistical significance of differences between groups. This is because the standard t-test assumes equal population variances. However, if this assumption does not hold, an adjusted form, such as Welch's t-test, is more appropriate. This modified test can accommodate unequal variances, offering more reliable results when variance homogeneity cannot be assured.