In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)


Readability of Font On a Computer Screen The statistics shown below were obtained from a standard test of readability of fonts on a computer screen (based on data from “Reading on the Computer Screen: Does Font Type Have Effects on Web Text Readability?” by Ali et al., International Education Studies, Vol. 6, No. 3). Reading speed and accuracy were combined into a readability performance score (x), where a higher score represents better font readability.


a. Use a 0.05 significance level to test the claim that there is no significant difference in readability between Roman and Arial fonts.

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that there is no significant difference in readability between Roman and Arial fonts, i.e., μ₁ = μ₂. The alternative hypothesis (H₁) states that there is a significant difference, i.e., μ₁ ≠ μ₂.
2. Step 2: Determine the test statistic formula. Since the population standard deviations are not assumed to be equal, use the two-sample t-test formula: t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂)), where x̄₁ and x̄₂ are the sample means, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes.
3. Step 3: Calculate the degrees of freedom (df). Use the formula for approximate degrees of freedom: df = ((s₁²/n₁ + s₂²/n₂)²) / (((s₁²/n₁)² / (n₁ - 1)) + ((s₂²/n₂)² / (n₂ - 1))). Round down to the nearest integer.
4. Step 4: Determine the critical t-value. Using the significance level α = 0.05 and the calculated degrees of freedom, find the critical t-value from a t-distribution table or technology. For a two-tailed test, divide α by 2 to find the critical values for both tails.
5. Step 5: Compare the calculated t-statistic to the critical t-value. If the absolute value of the t-statistic exceeds the critical t-value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis and conclude that there is no significant difference in readability between Roman and Arial fonts.

Key Concepts

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In this context, the Roman and Arial font readability scores are derived from separate groups of participants, meaning the performance of one group does not influence the other. This independence is crucial for applying certain statistical tests, such as the t-test, to compare the means of the two groups.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this case, the null hypothesis states that there is no significant difference in readability scores between the Roman and Arial fonts. The significance level of 0.05 indicates the threshold for determining whether the observed differences are statistically significant.

T-test for Independent Samples

The t-test for independent samples is a statistical test used to compare the means of two independent groups. It calculates the t-statistic based on the sample means, standard deviations, and sizes. Given that the population standard deviations are not assumed to be equal, the appropriate version of the t-test (Welch's t-test) should be used to account for the differences in variability between the two samples.