State what the given confidence interval suggests about the two population means.
-A hypothesis test is to be performed to test the equality of two population means. The sample sizes are large and the samples are independent. Give an expression for the population standard deviation of the $\left( \bar { x } _ { 1 } - \bar { x } _ { 2 } \right)$ values in terms of $\mathrm { s } _ { 1 } , \mathrm {~s} _ { 2 } , \mathrm { n } _ { 1 } \text {, and } \mathrm { n } _ { 2 }$
A) $\sqrt { \frac { s _ { 1 } ^ { 2 } } { n _ { 1 } } + \frac { s _ { 2 } ^ { 2 } } { n _ { 2 } } }$
B) $\sqrt { \frac { s ^ { 2 } } { 1 } }$
C) $\sqrt { \frac { \mathrm { s } _ { 1 } ^ { 2 } } { \mathrm { n } _ { 1 } } - \frac { \mathrm { s } _ { 2 } ^ { 2 } } { \mathrm { n } _ { 2 } } }$
D) $\frac { \mathrm { s } _ { 1 } ^ { 2 } } { \mathrm { n } _ { 1 } } + \frac { \mathrm { s } _ { 2 } ^ { 2 } } { \mathrm { n } _ { 2 } }$

E) $\frac { 1 } { n } + \frac { s } { n _ { 2 } }$

Question

State what the given confidence interval suggests about the two population means.
-A hypothesis test is to be performed to test the equality of two population means. The sample sizes are large and the samples are independent. Give an expression for the population standard deviation of the $\left( \bar { x } _ { 1 } - \bar { x } _ { 2 } \right)$ values in terms of $\mathrm { s } _ { 1 } , \mathrm {~s} _ { 2 } , \mathrm { n } _ { 1 } \text {, and } \mathrm { n } _ { 2 }$ 
A) $\sqrt { \frac { s _ { 1 } ^ { 2 } } { n _ { 1 } } + \frac { s _ { 2 } ^ { 2 } } { n _ { 2 } } }$
B) $\sqrt { \frac { s ^ { 2 } } { 1 } }$
C) $\sqrt { \frac { \mathrm { s } _ { 1 } ^ { 2 } } { \mathrm { n } _ { 1 } } - \frac { \mathrm { s } _ { 2 } ^ { 2 } } { \mathrm { n } _ { 2 } } }$
D) $\frac { \mathrm { s } _ { 1 } ^ { 2 } } { \mathrm { n } _ { 1 } } + \frac { \mathrm { s } _ { 2 } ^ { 2 } } { \mathrm { n } _ { 2 } }$

E) $\frac { 1 } { n } + \frac { s } { n _ { 2 } }$

Quizplus · Accepted Answer

The given confidence interval suggests that it is likely that the true difference between the population means falls between the lower and upper bounds given by the interval. Specifically, the interval is centered at the sample difference between the means and it is wide enough to account for the estimated variability of the sample means. Since the sample sizes are large and the samples are independent, it is reasonable to assume that the sampling distribution of $\left(\bar{x}_{1}-\bar{x}_{2}\right)$ is approximately normal. Therefore, the expression for the population standard deviation of $\left(\bar{x}_{1}-\bar{x}_{2}\right)$ in terms of $s_{1}$, $s_{2}$, $n_{1}$, and $n_{2}$ is given by option A: $\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}$. This reflects the fact that the variability of the difference between the means depends on the standard deviations of the population and the sample sizes. Option B is incorrect because it assumes that the standard deviations of the populations are equal, which is not given. Option C is incorrect because it has a negative sign, which is not possible in the standard deviation formula. Option D is incorrect because it adds the variances instead of taking their square root. Option E is incorrect because it has a term $\frac{1}{n}$ that does not belong in the formula for the standard deviation of the difference between two means.

State What the Given Confidence Interval Suggests About the Two (xˉ1−xˉ2)\left( \bar { x } _ { 1 } - \bar { x } _ { 2 } \right)(xˉ1​−xˉ2​)

State What the Given Confidence Interval Suggests About the Two $\left( \bar { x } _ { 1 } - \bar { x } _ { 2 } \right)$