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 } = \mu _ { 2 } ; \alpha = 0.10 \\). assume \\( \sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \\)
sample statistics: \\( \overline { x } _ { 1 } = 35.8, s _ { 1 } = 3.6, n _ { 1 } = 12 \\) and
\\( \overline { x } _ { 2 } = 37.8, s _ { 2 } = 2.4, n _ { 2 } = 16 \\)

identify the null and alternative hypotheses. choose the correct answer below.

a. \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
b. \\( h _ { 0 } : \mu _ { 1 } \geq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } < \mu _ { 2 } \\)
c. \\( h _ { 0 } : \mu _ { 1 } > \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \leq \mu _ { 2 } \\)
d. \\( h _ { 0 } : \mu _ { 1 } < \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \geq \mu _ { 2 } \\)
e. \\( h _ { 0 } : \mu _ { 1 } \leq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } > \mu _ { 2 } \\)
f. \\( h _ { 0 } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)

find the standardized test statistic t.
\\( t = - 1.76 \\)
(round to two decimal places as needed)

find the p - value.
\\( p = \\)
(round to three decimal places as needed.)

Explanation:

Step1: Determine the degrees of freedom

The formula for degrees of freedom when \(\sigma_1^{2}=\sigma_2^{2}\) is \(df=n_1 + n_2-2\).
Substitute \(n_1 = 12\) and \(n_2=16\) into the formula:
\(df=12 + 16-2=26\)

Step2: Calculate the P - value

Since the test is two - tailed (\(H_a:\mu_1
eq\mu_2\)) and \(t=-1.76\), we use the t - distribution table or a calculator.
Using a calculator (e.g., in R: \(2*pt(-1.76,26)\)), we find the P - value.
\(P = 2\times P(t_{26}\lt - 1.76)\)
\(P\approx0.090\)

Answer:

\(P = 0.090\)