Question

test the claim about the difference between two population means \\( \mu _ { 1 } \\) and \\( \mu _ { 2 } \\) at the level of significance \\( \alpha \\). assume th and independent, and the populations are normally distributed.
claim: \\( \mu _ { 1 } = \mu _ { 2 } ; \alpha = 0.01 \\)
population parameters: \\( \sigma _ { 1 } = 3.3, \sigma _ { 2 } = 1.5 \\)
sample statistics: \\( \overline { x } _ { 1 } = 16, n _ { 1 } = 28, \overline { x } _ { 2 } = 18, n _ { 2 } = 30 \\)
determine the alternative hypothesis.
\\( h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
determine the standardized test statistic.
\\( z = - 2.94 \\) (round to two decimal places as needed.)
determine the p - value.
\\( p - value = \square \\) (round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for P - value in a two - tailed z - test

For a two - tailed z - test with test statistic \(z\), the \(P-\text{value}=2\times(1 - \Phi(|z|))\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.

Step2: Calculate \(|z|\)

Given \(z=- 2.94\), then \(|z| = 2.94\).

Step3: Find \(\Phi(2.94)\)

Using a standard normal table or a calculator with a normal - distribution function (e.g., in Excel: NORM.S.DIST(2.94, TRUE)), we find that \(\Phi(2.94)\approx0.9984\).

Step4: Calculate the \(P-\text{value}\)

\(P-\text{value}=2\times(1 - 0.9984)=2\times0.0016 = 0.0032\approx0.003\)

Answer:

\(0.003\)