Suppose you want to test the claim that $\mu _ { 1 } = \mu _ { 2 }$ Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that $$\sigma { } _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 }$$ At a
Level of significance of α $\alpha$ = 0.01, when should you reject H₀? $$\begin{array} { l l }
\mathrm { n } _ { 1 } = 25 & \mathrm { n } _ { 2 } = 30 \
\overline { \mathrm { x } } 1 = 23 & \overline { \mathrm { x } } 2 = 21 \
\mathrm {~s} _ { 1 } = 1.5 & \mathrm {~s} _ { 2 } = 1.9
\end{array}$$
A)Reject H₀ if the standardized test statistic is less than -2.797 or greater than 2.797.
B)Reject H₀ if the standardized test statistic is less than -2.789 or greater than 2.797.
C)Reject H₀ if the standardized test statistic is less than -1.711 or greater than 1.711.
D)Reject H₀ if the standardized test statistic is less than -2.492 or greater than 2.492.

Question

Suppose you want to test the claim that $\mu _ { 1 } = \mu _ { 2 }$ Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that $$\sigma { } _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 }$$ At a
Level of significance of α $\alpha$ = 0.01, when should you reject H₀? $$\begin{array} { l l } 
\mathrm { n } _ { 1 } = 25 & \mathrm { n } _ { 2 } = 30 \
\overline { \mathrm { x } } 1 = 23 & \overline { \mathrm { x } } 2 = 21 \
\mathrm {~s} _ { 1 } = 1.5 & \mathrm {~s} _ { 2 } = 1.9
\end{array}$$
A)Reject H₀ if the standardized test statistic is less than -2.797 or greater than 2.797.
B)Reject H₀ if the standardized test statistic is less than -2.789 or greater than 2.797.
C)Reject H₀ if the standardized test statistic is less than -1.711 or greater than 1.711.
D)Reject H₀ if the standardized test statistic is less than -2.492 or greater than 2.492.

Quizplus · Accepted Answer

Given the unequal variances ($\sigma_1^2 \neq \sigma_2^2$), we use the Welch's t-test, which does not assume equal population variances. The degrees of freedom for this test are calculated using the sample sizes, means, and standard deviations, leading to a t-distribution critical value. At a significance level of $\alpha = 0.01$ (two-tailed), the critical values for rejecting the null hypothesis ($H_0: \mu_1 = \mu_2$) are typically found in t-distribution tables or calculated using statistical software. Given the provided options and the specific conditions (sample sizes, means, standard deviations, and $\alpha$), option A provides critical values that are reasonable for a two-tailed test at this significance level, considering the degrees of freedom would be adjusted due to the Welch's approximation.

Suppose You Want to Test the Claim That μ1=μ2\mu _ { 1 } = \mu _ { 2 }μ1​=μ2​

Suppose You Want to Test the Claim That $\mu _ { 1 } = \mu _ { 2 }$