Question

provided below are summary statistics for independent simple random samples from two populations. use the pooled t - test and the pooled t - interval procedure to conduct the required hypothesis test and obtain the specified confidence interval.
\\( \overline { x } _ { 1 } = 10, s _ { 1 } = 2.1, n _ { 1 } = 13, \overline { x } _ { 2 } = 12, s _ { 2 } = 2.1, n _ { 2 } = 13 \\)
a. two - tailed test, \\( \alpha = 0.01 \\)
b. 99% confidence interval
a. first, what are the correct hypotheses for a two - tailed test?
\\( \bigcirc a \\) \\( h _ { 0 } : \mu _ { 1 } < \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( \bigcirc b \\) \\( h _ { 0 } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( \bigcirc c \\) \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
\\( \bigcirc d \\) \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } > \mu _ { 2 } \\)
\\( \bigcirc e \\) \\( h _ { 0 } : \mu _ { 1 } > \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( \bigcirc f \\) \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } < \mu _ { 2 } \\)
next, compute the test statistic.
\\( t = - 2.428 \\) (round to three decimal places as needed.)
finally, determine the p - value.
\\( p = \square \\) (round to four decimal places as needed.)

Explanation:

Step1: Calculate the degrees of freedom

The degrees of freedom \( df=n_1 + n_2-2\). Given \(n_1 = 13\) and \(n_2 = 13\), so \(df=13 + 13-2=24\).

Step2: Determine the form of the P - value

Since it is a two - tailed test, \(P = 2P(t>\vert t_{stat}\vert)\) where \(t_{stat}=- 2.428\) (the absolute value \(\vert t_{stat}\vert = 2.428\)).
Using a t - distribution table or a calculator with a t - distribution function (e.g., in R: \(2(1 - pt(2.428,24))\) or in Excel: \(2(1 - T.DIST(2.428,24,TRUE))\)).

Step3: Calculate the P - value

\(P = 2\times(1 - 0.9901)=2\times0.0099 = 0.0198\)

Answer:

\(P = 0.0198\)