Construct a 95% confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ Two samples are random, independent, and come from populations that are normally distributed The sample statistics are given below. Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ $$\begin{array} { l l }
\mathrm { n } _ { 1 } = 11 & \mathrm { n } _ { 2 } = 18 \
\overline { \mathrm { x } } 1 = 4.8 & \overline { \mathrm { x } } 2 = 5.2 \
\mathrm {~s} _ { 1 } = 0.76 & \mathrm {~s} _ { 2 } = 0.51
\end{array}$$
A)(-0.883, 0.083)
B)(-4.152, 3.981)
C)(-2.762, 2.762)
D)(-1.762, 1.762)

Question

Construct a 95% confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ Two samples are random, independent, and come from populations that are normally distributed The sample statistics are given below. Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ $$\begin{array} { l l } 
\mathrm { n } _ { 1 } = 11 & \mathrm { n } _ { 2 } = 18 \
\overline { \mathrm { x } } 1 = 4.8 & \overline { \mathrm { x } } 2 = 5.2 \
\mathrm {~s} _ { 1 } = 0.76 & \mathrm {~s} _ { 2 } = 0.51
\end{array}$$
A)(-0.883, 0.083)
B)(-4.152, 3.981)
C)(-2.762, 2.762)
D)(-1.762, 1.762)

Quizplus · Accepted Answer

The correct confidence interval for $\mu_1 - \mu_2$ is calculated using the formula for two independent samples assuming equal variances. The formula for the confidence interval is $\overline{x}_1 - \overline{x}_2 \pm t_{\alpha/2, df} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$, where $t_{\alpha/2, df}$ is the t-value from the t-distribution for a 95% confidence level and degrees of freedom $df$, which can be approximated using the smaller of $n_1-1$ and $n_2-1$ or a pooled estimate. Given the sample sizes, means, and standard deviations, and assuming equal population variances, the calculation yields a confidence interval that closely matches choice A, indicating a range of plausible values for the difference in population means.

Construct a 95% Confidence Interval For μ1−μ2\mu _ { 1 } - \mu _ { 2 }μ1​−μ2​

Construct a 95% Confidence Interval For $\mu _ { 1 } - \mu _ { 2 }$