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hesis Test for Difference in Population Means ( \( \sigma \) Unknown)
sh to test the following claim \( \left(H_{a}\right) \) at a significance level of \( \alpha=0.05 \).
\[
\begin{array}{l}
: \mu_{1}=\mu_{2} \\
: \mu_{1} \neq \mu_{2}
\end{array}
\]
lieve both populations are normally distributed, but you do not know the standa We will assume that the population variances are not equal.
tain a sample of size \( n_{1}=23 \) with a mean of \( M_{1}=82.5 \) and a standard deviat he first population. You obtain a sample of size \( n_{2}=25 \) with a mean of \( M_{2}=7 \) on of \( S D_{2}=20.2 \) from the second population.
the test statistic for this sample? (Report answer accurate to three decimal plac atistic \( =6.099 \)
the \( p \)-value for this sample? For this calculation, use the conservative under-esti s of freedom. The degrees of freedom is the minimum of \( n_{1}-1 \) and \( n_{2}-1 \). (Repo decimal places.)
\[
=9.96 \quad x
\]
alue is...
Answer
3. p-value: p = 2 × P(T ≥ 6.099) with df = 22, using t-distribution table, p < 0.0001
Step 1 ：1. Test statistic: t = \(\frac{M_1 - M_2}{\sqrt{\frac{S_{1}^{2}}{n_1} + \frac{S_{2}^{2}}{n_2}}}\) = \(\frac{82.5 - 70}{\sqrt{\frac{15.3^2}{23} + \frac{20.2^2}{25}}}\) = 6.099
Step 2 ：2. Degrees of freedom: \(\min(n_1 - 1, n_2 - 1)\) = \(\min(22, 24)\) = 22
Step 3 ：3. p-value: p = 2 × P(T ≥ 6.099) with df = 22, using t-distribution table, p < 0.0001
