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Himanshu Kulshreshtha
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Asked: 2024-01-21T16:03:07+05:30

Differentiate between\(\mathrm{t}\) test and \(\mathrm{f}\) test.

Differentiate between\(\mathrm{t}\) test and \(\mathrm{f}\) test.

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    1. Elite Author
      Criteria t-Test F-Test (Analysis of Variance, ANOVA)
      Purpose Used to determine if there is a significant difference between the means of two independent groups or conditions. Used to assess whether there are significant differences in means among three or more independent groups or conditions.
      Types Independent samples t-test (two groups with independent data), paired samples t-test (two related groups with dependent data). One-way ANOVA (for comparing means of three or more groups), Two-way ANOVA (for assessing the influence of two independent factors).
      Assumptions Assumes that the data are normally distributed, populations have equal variances (homoscedasticity), and observations are independent. Assumes normality and homoscedasticity within each group. The data should also be independent between groups.
      Test Statistic Uses t-statistic to assess the difference between group means. Uses F-statistic to compare variance between group means and variance within groups.
      Hypotheses Null hypothesis ((H_0)): There is no significant difference between the group means. Alternative hypothesis ((H_1)): There is a significant difference between the group means. Null hypothesis ((H_0)): There is no significant difference between the group means. Alternative hypothesis ((H_1)): There is at least one group mean that is significantly different from the others.
      Degrees of Freedom df (degrees of freedom) depends on the specific t-test being used. For independent samples t-test, df = (n_1 + n_2 – 2) (where (n_1) and (n_2) are sample sizes). Two degrees of freedom: df1 (between-groups degrees of freedom) and df2 (within-groups degrees of freedom). df1 = k – 1 (where k is the number of groups), df2 = N – k (where N is the total number of observations).
      Test Statistic Formula (t = \frac{{\bar{x}_1 – \bar{x}_2}}{{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}}) (for independent samples t-test). (F = \frac{{\text{Between-Groups Variance}}}{\text{Within-Groups Variance}})
      P-Value Interpretation If p-value is less than the chosen significance level (e.g., 0.05), the null hypothesis is rejected, indicating a significant difference. If p-value is less than the chosen significance level, the null hypothesis is rejected, indicating at least one group mean is significantly different from others. Post hoc tests may be needed to identify specific group differences.
      Use Cases Commonly used when comparing means of two groups, such as before and after treatment or control vs. experimental groups. Appropriate for comparing means of three or more groups, such as in experimental designs with multiple conditions or groups.
      Example Comparing the mean test scores of students who received two different teaching methods. Comparing the mean scores of three different groups of participants who received three different types of therapy.
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