Let's solve the given problem step-by-step.
The formula for the [tex]\( t \)[/tex]-value is:
[tex]\[
t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}}
\][/tex]
Given values:
[tex]\[
\bar{x}_1 = 28.4646, \quad \bar{x}_2 = 25.1868, \quad \mu_1 - \mu_2 = 0, \quad s_p = 46.84, \quad n_1 = 47, \quad n_2 = 48
\][/tex]
### Step 1: Calculate the numerator of the [tex]\( t \)[/tex]-value formula
The numerator is:
[tex]\[
(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)
\][/tex]
Plugging in the given values:
[tex]\[
(28.4646 - 25.1868) - 0 = 3.2778
\][/tex]
### Step 2: Calculate the denominator of the [tex]\( t \)[/tex]-value formula
The denominator is:
[tex]\[
\sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}
\][/tex]
First, calculate [tex]\( s_p^2 \)[/tex]:
[tex]\[
s_p^2 = (46.84)^2 = 2194.5856
\][/tex]
Next, calculate the individual terms inside the square root:
[tex]\[
\frac{2194.5856}{47} \approx 46.6934
\][/tex]
[tex]\[
\frac{2194.5856}{48} \approx 45.7205
\][/tex]
Add these two values:
[tex]\[
46.6934 + 45.7205 \approx 92.4139
\][/tex]
Now take the square root of the sum:
[tex]\[
\sqrt{92.4139} \approx 9.6119
\][/tex]
### Step 3: Calculate the [tex]\( t \)[/tex]-value
Combine the numerator and the denominator:
[tex]\[
t = \frac{3.2778}{9.6119} \approx 0.3411
\][/tex]
### Step 4: Round the [tex]\( t \)[/tex]-value to two decimal places
[tex]\[
t \approx 0.34
\][/tex]
So, the [tex]\( t \)[/tex]-value is:
[tex]\[
t = 0.34
\][/tex]
