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  3. Intro to Statistics with R: Student's T-test

Exercise

Perform an independent t-test

In this exercise, you'll manually perform an independent t-test the same way you did for the dependent t-test in the previous chapter. Continuing with the working memory example, our null hypothesis is that the difference in intelligence score gain between the group that trained for 8 days and the group that trained for 19 days is equal to zero. If our observed t-value is sufficiently large, we can reject the null in favor of the alternative hypothesis, which would imply a significant difference in intelligence gain between the two training groups.

Calculation of the observed t-value for an independent t-test is similar to the dependent t-test, but involves slightly different formulas. The t-value is now

$$t = \frac{(\bar{x_1} - \bar{x_2})}{se_p}$$

where \(\bar{x_1}\) and \(\bar{x_1}\) are the mean intelligence gains for group 1 and group 2, respectively. \(se_p\) is the pooled standard error, which is equivalent to

$$se_p = \sqrt{ \frac{var_1}{n_1} + \frac{var_2}{n_2} }$$

where \(var_1\) and \(var_2\) are the variances and \(n_1\) and \(n_2\) are the sample sizes of groups 1 and 2, respectively.

Instructions

100 XP

The subsets of data for both 8 and 19 days of training, wm_t08 and wm_t19, are available in your workspace. Recall that the gain column contains the gain in intelligence score before and after training for each subject.

  • Compute the mean intelligence score gain for each of the two training groups and store the results in mean_t08 and mean_t19. Use the mean() function to do this.
  • Use the objects mean_t08 and mean_t19 to find the difference in means. Subtract the lowest mean from the highest mean and store the result in mean_diff.
  • Determine the number of participants in each sample using the code provided for you. Nothing to change here.
  • Calculate the degrees of freedom for the relevant t-distribution. The formula for degrees of freedom in an independent t-test is $n1 + n2 - 2$. Use the sample sizes you created above and save the result to df.
  • Create var_t08 and var_t19 by applying the var() function to the intelligence gain for each of the two training groups. These objects represent the variance for each group.
  • Compute the pooled standard error se_pooled using the formula outlined above. You will need to use the sqrt() function in addition to the other variables you've created in this exercise. Mind your brackets!