Does the Treatment Affect Success? The following table lists frequencies of successes and failures for different treatments used for a stress fracture in a foot bone (based on data from “Surgery Unfounded for Tarsal Navicular Stress Fracture,” by Bruce Jancin, Internal Medicine News, Vol. 42, No. 14). Use a 0.05 significance level to test the claim that success of the treatment is independent of the type of treatment. What does the result indicate about the increasing trend to use surgery?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to test whether the success of the treatment is independent of the type of treatment. This involves performing a chi-square test for independence, which is used to determine if there is a significant association between two categorical variables.
2. Step 2: Organize the data into a contingency table. The table should display the frequencies of successes and failures for each type of treatment. Label the rows as 'Type of Treatment' and the columns as 'Success' and 'Failure'.
3. Step 3: State the null and alternative hypotheses. The null hypothesis (H₀) is that the success of the treatment is independent of the type of treatment. The alternative hypothesis (H₁) is that the success of the treatment is not independent of the type of treatment.
4. Step 4: Calculate the expected frequencies for each cell in the contingency table using the formula: $E=\frac{R\times C}{N}$, where E is the expected frequency, R is the row total, C is the column total, and N is the grand total.
5. Step 5: Compute the chi-square test statistic using the formula: $\chi ²=\sum _i^j\frac{(O-E)²}{E}$, where O is the observed frequency and E is the expected frequency. Compare the test statistic to the critical value from the chi-square distribution table at the 0.05 significance level to determine whether to reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that assumes no effect or relationship, and an alternative hypothesis (H1) that suggests a significant effect or relationship. The goal is to determine whether the observed data provides enough evidence to reject the null hypothesis at a specified significance level, such as 0.05.

Chi-Square Test of Independence

The Chi-Square Test of Independence is a statistical test used to determine if there is a significant association between two categorical variables. In this context, it assesses whether the success of a treatment is independent of the type of treatment used. The test compares the observed frequencies of outcomes in a contingency table to the expected frequencies if the variables were independent, providing a p-value to evaluate significance.

Significance Level

The significance level, often denoted as alpha (α), is the threshold used to determine whether to reject the null hypothesis in hypothesis testing. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none (Type I error). If the p-value obtained from the test is less than or equal to the significance level, the null hypothesis is rejected, suggesting a statistically significant effect or relationship.