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The paper "If It's Hard to Read, It's Hard to Do" (Psychological Science [2008]\(: 986-988)\) 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. $$ \begin{array}{ccc} & \text { Easy font } & \text { Difficult font } \\ \hline n & 10 & 10 \\ \bar{x} & 8.23 & 15.10 \\ s & 5.61 & 9.28 \\ \hline \end{array} $$ 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

Understanding a two-sample t test

A two-sample t test is a hypothesis test that compares the means of two populations. The test is used to determine whether there is significant evidence that the means are different. The primary assumptions for this test are: Independence of observations within and between groups, approximately normally distributed responses within each group, and equal variances of the two populations. The first assumption seems to be met since individuals were randomly assigned to the two groups.

02

Checking assumptions

For the second assumption, normal distribution of response within each group, we don't have enough information. The standard deviations are provided but we can't make an inference about normality without a plot of the data or statistical measures of skewness and kurtosis. The third assumption regarding equal variances is also hard to determine with certainty given only summary statistics. Though from the provided data, the standard deviation of the difficult font group is higher than that of the easy font group.

03

Conclusion

Therefore, despite a t test being possibly the correct statistical approach for comparing means of two independent samples, in this case, it might not be entirely appropriate due to potential violation of the assumptions of equal variances and normality. Without additional information, it's difficult to judge if it is permissible to use a two-sample t test in this situation.

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Key Concepts

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

Normal Distribution

The concept of normal distribution is a central pillar in statistics, especially in hypothesis testing. A normal distribution is a symmetric, bell-shaped curve that characterizes the spread of a set of data. It is vital because much of statistical theory is based on this model.

In the context of a two-sample t test, the assumption is that the data from each group being compared are normally distributed. This means that within each group, if you were to plot the data, it should resemble a bell-shaped curve. However, it's important to note that without concrete data—like a histogram or numerical tests like the Shapiro-Wilk test—verifying normality based simply on summary statistics can be tricky.
- Normal distribution allows us to make probabilistic statements about data - Many tests, including the t test, assume normality for reliability
The study mentioned in the exercise assumes normal distribution for both easy and difficult font groups' responses. Without specific evidence of normality from the data, caution should be exercised when interpreting results.

Independent Samples

Independent samples mean that the performance or outcome of one group does not affect or relate to the other group. In experiments, ensuring independence is crucial to maintaining the validity of the conclusions.

The study in question used random assignment to assign students to either the easy font group or the difficult font group. This randomization process is intended to ensure that the groups are independent of each other.
- Independence supports the validity of comparative statistical tests - Random assignment is a common technique to achieve independence
For the two-sample t test, the assumption of independent samples is key. If randomization was properly followed, then this assumption is likely met. However, external factors or biases not controlled for could potentially violate independence.

Equal Variance Assumption

The equal variance assumption states that the variance in outcomes should be the same or similar across the groups being compared. Variance is a measure of how data points differ from the mean.

In a two-sample t test, if the variances are not equal (known as heteroscedasticity), the test results may not be valid. The exercise provides standard deviations of 5.61 for the easy font group and 9.28 for the difficult font group. This suggests potentially different variances.
- Equal variances ensure reliability in the comparison of means - Standard deviations offer a clue but not proof of equal variances
To check for this, a test such as Levene's test for equality of variances could be used. In the absence of such tests, one must be cautious in interpreting t test results if variances seem disparate from summary data alone.

Hypothesis Testing

Hypothesis testing is a procedural method used in statistics to test a claim or theory about a population parameter. It involves comparing an experimental group to a control group to determine statistical significance.

In the context of the exercise, a two-sample t test was used to test the hypothesis whether the average estimated time to complete a task was different for groups receiving easy-to-read and difficult-to-read instructions. The hypotheses set were:

• Null Hypothesis ( H_0 ): Mean times are equal.
• Alternative Hypothesis ( H_a ): Mean time for the easy font is less than for the difficult font.

- The t test evaluates if observed data deviate from the null hypothesis to a degree that is statistically significant - A significance level of 0.10 implies a 10% risk of concluding a difference exists when it doesn't
If the p-value obtained from the test is less than the significance level, the null hypothesis can be rejected in favor of the alternative. But given the importance of the other assumptions, such as normal distribution and equal variance, the conclusions drawn from such a test must be carefully considered.