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? $$\begin{array} { l l }
\mathrm { n } _ { 1 } = 18 & \mathrm { n } _ { 2 } = 13 \
\overline { \mathrm { x } }_ 1 = 595 & \overline { \mathrm { x } }_ 2 = 580 \
\mathrm {~s} 1 = 40 & \mathrm {~s} 2 = 25
\end{array}$$
A)Reject H₀ if the standardized test statistic is greater than 2.681.
B)Reject H₀ if the standardized test statistic is greater than 1.699.
C)Reject H₀ if the standardized test statistic is greater than 2.179.
D)Reject H₀ if the standardized test statistic is greater than 3.055.

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? $$\begin{array} { l l } 
\mathrm { n } _ { 1 } = 18 & \mathrm { n } _ { 2 } = 13 \
\overline { \mathrm { x } }_ 1 = 595 & \overline { \mathrm { x } }_ 2 = 580 \
\mathrm {~s} 1 = 40 & \mathrm {~s} 2 = 25
\end{array}$$
A)Reject H₀ if the standardized test statistic is greater than 2.681.
B)Reject H₀ if the standardized test statistic is greater than 1.699.
C)Reject H₀ if the standardized test statistic is greater than 2.179.
D)Reject H₀ if the standardized test statistic is greater than 3.055.

Quizplus · Accepted Answer

Given the unequal variances ($\sigma_1^2 \neq \sigma_2^2$), and the sample sizes along with the level of significance ($\alpha = 0.01$), the correct approach is to use the Welch's t-test. The degrees of freedom for this test are calculated using the sample sizes and standard deviations, leading to a t-distribution that does not assume equal variances. The critical value for a one-tailed test at $\alpha = 0.01$ with degrees of freedom that would approximate those given by the Welch-Satterthwaite equation (since exact degrees of freedom are not provided, we infer based on typical values for such sample sizes and variance conditions) is closest to the value provided in option A, which is for more conservative tests typically used in situations with unequal variances and smaller sample sizes.

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

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