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 = 0.01 \\)
population parameters: \\( \sigma _ { 1 } = 3.6, \sigma _ { 2 } = 1.4 \\)
sample statistics: \\( \overline { x } _ { 1 } = 17, n _ { 1 } = 30, \overline { x } _ { 2 } = 15, n _ { 2 } = 26 \\)
determine the alternative hypothesis.
\\( h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
determine the standardized test statistic.
\\( z = 2.81 \\) (round to two decimal places as needed)
determine the p - value.
\\( p - value = \square \\) (round to three decimal places as needed)

Explanation:

Step1: Determine the formula for P - value

Since the alternative hypothesis is \(H_a:\mu_1
eq\mu_2\), this is a two - tailed test. The formula for the P - value in a two - tailed z - test is \(P - value = 2\times(1 - P(Z<|z|))\), where \(z\) is the standardized test statistic.

Step2: Find the probability for the given z - score

We know \(z = 2.81\). Using a standard normal table or a calculator with a normal distribution function (e.g., in Excel: =NORM.S.DIST(2.81,TRUE)), we find that \(P(Z < 2.81)\approx0.9975\)

Step3: Calculate the P - value

Substitute into the formula: \(P - value=2\times(1 - 0.9975)\)

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Answer:

\(0.005\)