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The paper "If It's Hard to Read, It's Hard to Do" (Psychological Science [20081: 986-988) described an interesting study of how people perceive the effort required to do certain tasks. Each of 20 students was randomly assigned to one of two groups. One group was given instructions for an exercise routine that were printed in an easy-to-read font (Arial). The other group received the same set of instructions, but printed in a font that is considered difficult to read (Brush). After reading the instructions, subjects estimated the time (in minutes) they thought it would take to complete the exercise routine. Summary statistics are given below. The authors of the paper used these data to carry out a two-sample \(t\) test, and concluded that at the 10 significance level, there was convincing evidence that the mean estimated time to complete the exercise routine was less when the instructions were printed in an easy-to-read font than when printed in a difficult-to-read font. Discuss the appropriateness of using a two-sample \(t\) test in this situation.

Step by step solution

01

Identifying the samples

First, identify the samples involved in the study. Here, there are two independent samples which consist of students who read the exercise routine instructions in easy-to-read and hard-to-read fonts.

02

Assessing normality

Next, assess whether the population distribution of estimated times to complete the exercise, for both the easy-to-read and difficult-to-read fonts, is roughly normal. Given that the sample sizes are relatively small, it can be challenging to definitively determine normality from the summaries alone. In a practical setting, a QQ plot or Histogram could be used to visually assess normality.

03

Assessing independence

Evaluate whether the estimated times for the two groups are independent. In this study, random assignment was used to form the groups, so we can assume that the responses are independent within and between groups.

04

Comparing variances

Lastly, you should compare the variances for the two groups. The two-sample t test assumes that the population variances are approximately equal. If this assumption is dramatically violated, it may impact the validity of the t test result. The exercise doesn't provide variance or standard deviation values which means we cannot make this assessment based on the provided information.

05

Making a conclusion

Based on the provided information, with the assumption that the distribution of time estimates are approximately normal, and the fact that independence seems justified as students were randomly assigned, it would be appropriate to use a two-sample t test in this case. However, the equal variance assumption was unable to be assessed which could potentially impact the conclusion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normality Assessment

A crucial aspect of conducting a two-sample t-test is the normality assessment. This means checking if the data for each group follows a normal distribution. In our exercise, this involves the estimated times to complete the exercise routine for both the easy-to-read and hard-to-read fonts. Since our samples are relatively small with only 20 participants, it's challenging to assert normality just based on summary statistics. Ideally, visual tools like QQ plots or histograms are used to observe the data's distribution shape. However, in a practical sense, if the sample sizes were larger, the central limit theorem might justify the assumption of normality, making the t-test results more reliable.

Independent Samples

It's essential to ensure that the samples used in the test are independent. This means the estimated times from each student in one group do not affect the times in the other group. In our study, this assumption holds true because random assignment was used to allocate each of the 20 students to the groups. Consequently, this random allocation helps in maintaining the independence of the samples, reducing any bias and ensuring that the responses between groups are unlinked.

Equal Variance Assumption

The equal variance assumption is another critical requirement for a two-sample t-test. This assumption suggests that the variances of the two populations should be approximately equal. In our exercise, we are not explicitly provided with the variances for the two font reading difficulty levels. Thus, we are unable to test for this assumption rigorously. Usually, statistical tests like Levene's test can be employed to check this assumption but, in our context, we can only proceed with a t-test by assuming inherent equal variances for a valid comparison.

Random Assignment

Random assignment plays a pivotal role in experimental studies. It involves randomly distributing participants across different groups to ensure that any differences observed between groups can be attributed to the treatment or condition applied, not pre-existing differences among participants. In this exercise, the use of random assignment helps to enhance the validity of the results. When students were assigned to read instructions in varied font difficulties at random, it ensures that each student's pre-existing characteristics (e.g., baseline reading ability) are evenly dispersed across both groups. This randomness aids in achieving unbiased and independent outcomes, enhancing the credibility of the findings analyzed with a two-sample t-test.