use the given statistics to complete parts (a) and (b). assume that the populations are normally distributed
(a) test whether $ \mu _ { 1 } \mu _ { 2 } $ at the $ \alpha = 0.01 $ level of significance for the given sample data
(b) construct a $ 99 \% $ confidence interval about $ \mu _ { 1 } - \mu _ { 2 } $
0.52 (round to two decimal places as needed)
determine the p - value for this hypothesis test.
0.305 (round to three decimal places as needed)
state the conclusion for this hypothesis test.
a. reject $ h _ { 0 } $. there is not sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } \mu _ { 2 } $
b. reject $ h _ { 0 } $. there is sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } \mu _ { 2 } $
c. do not reject $ h _ { 0 } $. there is sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } \mu _ { 2 } $
d. do not reject $ h _ { 0 } $. there is not sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } \mu _ { 2 } $
(b) the $ 99 \% $ confidence interval about $ \mu _ { 1 } - \mu _ { 2 } $ is the range from a lower bound of $ \square $ to an upper bound of $ \square $
(round to three decimal places as needed.)

Question

use the given statistics to complete parts (a) and (b). assume that the populations are normally distributed
(a) test whether $ \mu _ { 1 } > \mu _ { 2 } $ at the $ \alpha = 0.01 $ level of significance for the given sample data
(b) construct a $ 99 \% $ confidence interval about $ \mu _ { 1 } - \mu _ { 2 } $
0.52 (round to two decimal places as needed)
determine the p - value for this hypothesis test.
0.305 (round to three decimal places as needed)
state the conclusion for this hypothesis test.
a. reject $ h _ { 0 } $. there is not sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } > \mu _ { 2 } $
b. reject $ h _ { 0 } $. there is sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } > \mu _ { 2 } $
c. do not reject $ h _ { 0 } $. there is sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } > \mu _ { 2 } $
d. do not reject $ h _ { 0 } $. there is not sufficient evidence at the $ \alpha = 0.01 $ level of significance to conclude that $ \mu _ { 1 } > \mu _ { 2 } $
(b) the $ 99 \% $ confidence interval about $ \mu _ { 1 } - \mu _ { 2 } $ is the range from a lower bound of $ \square $ to an upper bound of $ \square $
(round to three decimal places as needed.)

Accepted Answer

# Explanation:
## Step1: Calculate the degrees of freedom
For two - sample $t$ - test, the degrees of freedom $df=\min(n_1 - 1,n_2 - 1)$. Here $n_1 = 26$, $n_2=18$. So $df=\min(26 - 1,18 - 1)=17$

## Step2: Determine the critical value for confidence interval
For a $99\%$ confidence interval and $df = 17$, the critical value $t_{\alpha/2}$ (where $\alpha=1 - 0.99 = 0.01$, $\alpha/2=0.005$) from the $t$ - distribution table is $t_{0.005,17}=2.898$

## Step3: Calculate the standard error
The formula for the standard error $SE=\sqrt{\frac{s_1^{2}}{n_1}+\frac{s_2^{2}}{n_2}}$, where $s_1 = 4.0$, $n_1 = 26$, $s_2=9.8$, $n_2 = 18$
$$
\begin{align*}
SE&=\sqrt{\frac{4.0^{2}}{26}+\frac{9.8^{2}}{18}}\
&=\sqrt{\frac{16}{26}+\frac{96.04}{18}}\
&=\sqrt{0.6154+5.3356}\
&=\sqrt{5.951}\
&\approx2.44
\end{align*}
$$

## Step4: Calculate the lower and upper bounds of the confidence interval
The point estimate is $\bar{x}_1-\bar{x}_2=45.7 - 44.4 = 1.3$
The lower bound $LB=(\bar{x}_1-\bar{x}_2)-t_{\alpha/2}\times SE$
$$
\begin{align*}
LB&=1.3-2.898\times2.44\
&=1.3 - 7.071\
&=- 5.771
\end{align*}
$$
The upper bound $UB=(\bar{x}_1-\bar{x}_2)+t_{\alpha/2}\times SE$
$$
\begin{align*}
UB&=1.3 + 2.898\times2.44\
&=1.3+7.071\
&=8.371
\end{align*}
$$

# Answer:
The $99\%$ confidence interval about $\mu_1-\mu_2$ is the range from a lower bound of $-5.771$ to an upper bound of $8.371$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the degrees of freedom

Step2: Determine the critical value for confidence interval

Step3: Calculate the standard error

Step4: Calculate the lower and upper bounds of the confidence interval

Answer: