The one-sample t-test is used to determine if the mean of a single sample is significantly different from a hypothesized population mean. For instance, suppose we want to test if the average height of students in a school is significantly different from the average height of people in general. We collect the heights of a sample of students and compare it to the known population mean height. If the p-value is less than the chosen significance level, we can reject the null hypothesis and conclude that there is a significant difference in the heights.
The paired t-test is used when we have paired or matched samples. For example, we can analyze the impact of a new teaching method by measuring students' scores before and after implementing the method. By comparing the mean differences between the pairs, we can determine if there is a significant improvement or not.
The independent two-sample t-test allows us to compare the means of two independent groups. For instance, we might want to know if there is a significant difference in the test scores of male and female students. We collect samples from each group, and the t-test helps us determine if the difference in means is statistically significant or simply due to random chance.
The chi-square (χ²) test is used to analyze the association between two categorical variables. It allows us to determine if there is a significant relationship or dependency between the variables. An example could be analyzing the association between gender and smoking habits among a group of individuals. By comparing the observed frequencies with the expected frequencies, we can make conclusions about the relationship.
Remember, each hypothesis test has its specific application and assumptions. Understanding the appropriate test for your research question will lead to more accurate and reliable results.