Preliminary data analyses indicate that use of a paired...

Preliminary data analyses indicate that use of a paired t-test is reasonable. Perform each hypothesis test by using either the critical-value approach or the P-value approach. Glaucoma and Corneal Thickness. Glaucoma is a leading cause of blindness in the United States. N. Fhlers measured the corneal thickness of eight patients who had glaucoma in one eye but not in the other. The results of the study were published as the paper "On Corneal Thickness and Intraocular Pressure, II" (Acta Ophthalmology, Vol. 48, pp. 1107-1112). The following are the data on cornmeal thickness, in microns. At the $10 \%$ significance level, do the data provide sufficient evidence to conclude that mean corneal thickness is greater in normal eyes than in eyes with glaucoma?

Preliminary data analyses indicate that use of a paired t-test is reasonable. Perform each hypothesis test by using either the critical-value approach or the P-value approach.
Glaucoma and Corneal Thickness. Glaucoma is a leading cause of blindness in the United States. N. Fhlers measured the corneal thickness of eight patients who had glaucoma in one eye but not in the other. The results of the study were published as the paper "On Corneal Thickness and Intraocular Pressure, II" (Acta Ophthalmology, Vol. 48, pp. 1107-1112). The following are the data on cornmeal thickness, in microns.
At the $10 \%$ significance level, do the data provide sufficient evidence to conclude that mean corneal thickness is greater in normal eyes than in eyes with glaucoma?

Key Concepts

Paired t-test

A paired t-test is a statistical method used to compare the means from two related groups. It accounts for the dependency between paired observations, which can arise when measurements are taken from the same subjects under different conditions. This test is particularly useful when the same subjects receive two different treatments or when matched subjects are compared to control for individual variability.

Hypothesis Testing

Hypothesis testing is a framework for making inferences about population parameters based on sample data. In this process, a null hypothesis (usually representing no effect or no difference) is tested against an alternative hypothesis. The outcome of the test determines whether there is enough statistical evidence to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) is a statement that there is no effect or no difference, while the alternative hypothesis (Ha) represents the outcome that the researcher expects or is trying to provide evidence for. The definitions of these hypotheses guide the testing procedure and the interpretation of the results.

Significance Level

The significance level, often denoted as alpha, is the probability threshold used to decide whether to reject the null hypothesis. It represents the maximum acceptable probability of making a Type I error, which is the error of rejecting a true null hypothesis. Common significance levels include 0.05, 0.01, and 0.10, depending on the context and desired strictness of the test.

Critical-Value Approach

The critical-value approach involves comparing the test statistic calculated from the sample data to a critical value derived from the appropriate statistical distribution under the null hypothesis. If the test statistic falls in the critical region (beyond the critical value), the null hypothesis is rejected. This method provides a straightforward decision rule based on predetermined criteria.

P-value Approach

The P-value approach involves calculating the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming that the null hypothesis is true. If the P-value is less than or equal to the significance level, the null hypothesis is rejected. This approach provides an exact measure of the statistical evidence against the null hypothesis.

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