Richilan Radials advertises that Richilan radial tires are...

Richilan Radials advertises that Richilan radial tires are better than Randy Radials. As proof, they tested 20 of their radials and 20 Randy Radials. Richilan claims that only 2 of their tires failed to reach 50,000 miles, but 7 Randy Radials failed. Use the Fisher-Irwin test at the $5 \%$ level of significance to test Richilan's claim. The null hypothesis is that there is no difference in the tires. The alternative hypothesis is that the fraction of good Richilan tires is higher than the fraction of good Randy tires.

Key Concepts

Fisher’s Exact Test

Fisher’s Exact Test (often called the Fisher-Irwin test in some texts) is a statistical significance test used in the analysis of contingency tables. It is particularly valuable when sample sizes are small and the assumptions of more approximate methods, such as the chi-square test, do not hold. The test computes the exact probability of obtaining a distribution of values in a 2×2 table, under the null hypothesis of no association between the groups.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis is the default assumption that there is no effect or difference, while the alternative hypothesis represents the claim to be tested. In this context, the null hypothesis posits no difference between the performance (good tire proportions) of two tire brands, and the alternative hypothesis asserts that one brand has a higher proportion of good outcomes compared to the other.

Two-Sample Proportion Comparison

Two-sample proportion comparison is a method used to determine whether there is a statistically significant difference between the proportions of a particular outcome in two independent groups. It is commonly applied in quality control, medical studies, and social sciences where proportions or probabilities between two distinct groups are compared.

Significance Level

The significance level is a predetermined threshold (often set at 5%) that defines the probability of rejecting the null hypothesis when it is actually true (Type I error). It dictates how extreme the test statistic must be before the null hypothesis is rejected, thereby controlling the likelihood of a false positive conclusion.

One-Tailed Test

A one-tailed test is used when the research hypothesis specifies the direction of the expected difference. Here, the alternative hypothesis is directional—claiming that the proportion of good outcomes for one group is higher than that for the other. This approach allocates the entire significance level to detecting an effect in that specific direction.

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