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 = 3.5 \\), \\( \sigma_2 = 1.7 \\)
sample statistics: \\( \bar{x}_1 = 15 \\), \\( n_1 = 29 \\), \\( \bar{x}_2 = 17 \\), \\( n_2 = 30 \\\\

determine the alternative hypothesis.
\\( h_a: \mu_1 \
eq \mu_2 \\\\
determine the standardized test statistic.
\\( z = -2.78 \\) (round to two decimal places as needed.)
determine the p - value.
p - value = \\( \square \\) (round to three decimal places as needed.)

Explanation:

Step1: Recall P - value formula for two - tailed test

For a two - tailed z - test, the P - value is calculated as \(P - value=2\times(1 - \Phi(|z|))\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution. We know that the test statistic \(z=- 2.78\), so \(|z| = 2.78\).

Step2: Find \(\Phi(2.78)\)

Using the standard normal table or a calculator with a normal distribution function, we find that \(\Phi(2.78)\approx0.9973\).

Step3: Calculate the P - value

Substitute the value of \(\Phi(2.78)\) into the formula for the P - value.
\(P - value = 2\times(1 - 0.9973)=2\times0.0027 = 0.0054\approx0.005\) (rounded to three decimal places)

Answer:

\(0.005\)