Math Snap

STEP 1

1. We are given the values: $\bar{x}_{1}=27.6106$, $\bar{x}_{2}=25.9487$, $\mu_{1}-\mu_{2}=0$, $s_{1}=43.83$, $s_{2}=43.83$, $n_{1}=40$, and $n_{2}=43$.
2. The formula to evaluate is:
$$ t=\frac{\left(\bar{x}_{1}-\bar{x}_{2}\right)-\left(\mu_{1}-\mu_{2}\right)}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}
\] 3. The result should be rounded to two decimal places.

STEP 2

1. Calculate the numerator of the formula.
2. Calculate the denominator of the formula.
3. Evaluate the formula and round the result.

STEP 3

Calculate the numerator:
$$\left(\bar{x}_{1}-\bar{x}_{2}\right) - \left(\mu_{1}-\mu_{2}\right) = (27.6106 - 25.9487) - 0 = 1.6619$$

STEP 4

Calculate the denominator:
$$\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}} = \sqrt{\frac{43.83^2}{40} + \frac{43.83^2}{43}}$$ Calculate each term:
$$\frac{43.83^2}{40} = \frac{1921.1889}{40} = 48.0297225$$ $$\frac{43.83^2}{43} = \frac{1921.1889}{43} = 44.6741628$$ Add the terms:
$$48.0297225 + 44.6741628 = 92.7038853$$ Take the square root:
$$\sqrt{92.7038853} \approx 9.6304$$

Evaluate the formula:
$$t = \frac{1.6619}{9.6304} \approx 0.1725$$ Round to two decimal places:
$$t \approx 0.17$$ The value of $t$ is:
$$\boxed{0.17}$$