Answer:
To calculate the t-statistic for independent samples, we use the formula:
\[ t = \frac{{\bar{x}_1 - \bar{x}_2}}{{\sqrt{\frac{{s_1^2}}{{n_1}} + \frac{{s_2^2}}{{n_2}}}}} \]
Where:
- \( \bar{x}_1 \) and \( \bar{x}_2 \) are the means of Group 1 and Group 2 respectively.
- \( s_1^2 \) and \( s_2^2 \) are the variances of Group 1 and Group 2 respectively.
- \( n_1 \) and \( n_2 \) are the sample sizes of Group 1 and Group 2 respectively.
Given:
- \( \bar{x}_1 = 81.81 \)
- \( \bar{x}_2 = 83.55 \)
- \( s_1^2 = s_2^2 = 4.2 \)
- \( n_1 = 20 \)
- \( n_2 = 21 \)
Let's plug in the values:
\[ t = \frac{{81.81 - 83.55}}{{\sqrt{\frac{{4.2}}{{20}} + \frac{{4.2}}{{21}}}}} \]
\[ t = \frac{{-1.74}}{{\sqrt{0.21 + 0.2}}} \]
\[ t = \frac{{-1.74}}{{\sqrt{0.41}}} \]
\[ t \approx \frac{{-1.74}}{{0.64}} \]
\[ t \approx -2.72 \]
Rounded to the nearest two decimal places, the t-statistic is approximately -2.72.