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 } = 31.4, s _ { 1 } = 3.4, n _ { 1 } = 11 \\) and
\\( \overline { x } _ { 2 } = 33.9, s _ { 2 } = 2.4, n _ { 2 } = 19 \\)
identify the null and alternative hypotheses. choose the correct answer below.
a. \\( h _ { 0 } : \mu _ { 1 } \leq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } > \mu _ { 2 } \\)
b. \\( h _ { 0 } : \mu _ { 1 } < \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \geq \mu _ { 2 } \\)
c. \\( h _ { 0 } : \mu _ { 1 } > \mu _ { 2 } \\)
d. \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \leq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
e. \\( h _ { 0 } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
f. \\( h _ { 0 } : \mu _ { 1 } \geq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } < \mu _ { 2 } \\)
find the standardized test statistic t.
\\( t = - 2.35 \\)
(round to two decimal places as needed.)
find the p - value.
\\( p = \square \\)
(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 \(n_1 = 11\) and \(n_2=19\) into the formula:
\(df=11 + 19-2=28\)

Step2: Find the P - value

Since the test is two - tailed (because \(H_a:\mu_1
eq\mu_2\)) and \(t=- 2.35\), we use the t - distribution table or a calculator.
Using a t - distribution calculator with \(df = 28\) and \(|t|=2.35\), the P - value is \(2P(t>2.35)\) (for a two - tailed test).
Using a calculator or software (e.g., in R: 2*(1 - pt(2.35,28))), we get \(P = 0.026\)

Answer:

\(P = 0.026\)