test the claim about the difference between two population means \$ \mu _ { 1 } \$ and \$ \mu _ { 2 } \$ at the level of significance \$ \alpha \$. assume the samples are random and independent, and the populations are normally distributed.
claim: \$ \mu _ { 1 } \leq \mu _ { 2 } ; \alpha = 0.10 \$. assume \$ \sigma _ { 1 } ^ { 2 } \
eq \sigma _ { 2 } ^ { 2 } \$
sample statistics: \$ \overline { x } _ { 1 } = 2409, s _ { 1 } = 174, n _ { 1 } = 14 \$ and
\$ \overline { x } _ { 2 } = 2300, s _ { 2 } = 53, n _ { 2 } = 10 \$
identify the null and alternative hypotheses. choose the correct answer below.
\$ \bigcirc \$ a. \$ h _ { 0 } : \mu _ { 1 } < \mu _ { 2 } \$
\$ h _ { a } : \mu _ { 1 } \geq \mu _ { 2 } \$
\$ \bigcirc \$ b. \$ h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \$
\$ h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \$
\$ \bigcirc \$ c. \$ h _ { 0 } : \mu _ { 1 } \leq \mu _ { 2 } \$
\$ h _ { a } : \mu _ { 1 } \gt \mu _ { 2 } \$
\$ \bigcirc \$ d. \$ h _ { 0 } : \mu _ { 1 } > \mu _ { 2 } \$
\$ h _ { a } : \mu _ { 1 } \leq \mu _ { 2 } \$
\$ \bigcirc \$ e. \$ h _ { 0 } : \mu _ { 1 } \
eq \mu _ { 2 } \$
\$ h _ { a } : \mu _ { 1 } = \mu _ { 2 } \$
\$ \bigcirc \$ f. \$ h _ { 0 } : \mu _ { 1 } \geq \mu _ { 2 } \$
\$ h _ { a } : \mu _ { 1 } < \mu _ { 2 } \$
find the standardized test statistic t.
\$ t = \$

Question

test the claim about the difference between two population means $ \mu _ { 1 } $ and $ \mu _ { 2 } $ at the level of significance $ \alpha $. assume the samples are random and independent, and the populations are normally distributed.
claim: $ \mu _ { 1 } \leq \mu _ { 2 } ; \alpha = 0.10 $. assume $ \sigma _ { 1 } ^ { 2 } \
eq \sigma _ { 2 } ^ { 2 } $
sample statistics: $ \overline { x } _ { 1 } = 2409, s _ { 1 } = 174, n _ { 1 } = 14 $ and
$ \overline { x } _ { 2 } = 2300, s _ { 2 } = 53, n _ { 2 } = 10 $
identify the null and alternative hypotheses. choose the correct answer below.
$ \bigcirc $ a. $ h _ { 0 } : \mu _ { 1 } < \mu _ { 2 } $
$ h _ { a } : \mu _ { 1 } \geq \mu _ { 2 } $
$ \bigcirc $ b. $ h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } $
$ h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } $
$ \bigcirc $ c. $ h _ { 0 } : \mu _ { 1 } \leq \mu _ { 2 } $
$ h _ { a } : \mu _ { 1 } \gt \mu _ { 2 } $
$ \bigcirc $ d. $ h _ { 0 } : \mu _ { 1 } > \mu _ { 2 } $
$ h _ { a } : \mu _ { 1 } \leq \mu _ { 2 } $
$ \bigcirc $ e. $ h _ { 0 } : \mu _ { 1 } \
eq \mu _ { 2 } $
$ h _ { a } : \mu _ { 1 } = \mu _ { 2 } $
$ \bigcirc $ f. $ h _ { 0 } : \mu _ { 1 } \geq \mu _ { 2 } $
$ h _ { a } : \mu _ { 1 } < \mu _ { 2 } $
find the standardized test statistic t.
$ t = $

Accepted Answer

# Explanation:
## Step1: Recall the formula for the t - statistic for two - sample t - test (unequal variances)
The formula for the t - statistic is $t=\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$

## Step2: Substitute the given values into the formula
We are given $\bar{x}_{1} = 2409$, $\bar{x}_{2}=2300$, $s_{1}=174$, $s_{2} = 53$, $n_{1}=14$, and $n_{2}=10$

First, calculate the numerator: $\bar{x}_{1}-\bar{x}_{2}=2409 - 2300=109$

Then, calculate the denominator:
$\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}=\frac{174^{2}}{14}+\frac{53^{2}}{10}$
$=\frac{30276}{14}+\frac{2809}{10}$
$ = 2162.5714+280.9$
$=2443.4714$
$\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}=\sqrt{2443.4714}\approx49.43$

Now, $t=\frac{109}{49.43}\approx2.205$

# Answer:
$t\approx2.21$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for the t - statistic for two - sample t - test (unequal variances)

Step2: Substitute the given values into the formula

Answer: