Hint
Recall the basic form of the z-test statistic for comparing means and the role of the standard error in its calculation. Ensure you address all steps of hypothesis testing.
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Complete Answer
To test for a significant difference between the means of two independent production lines, we follow these steps:
First, formulate the null hypothesis (\(H_0\)) that there is no difference between the population means (\(\mu_1 = \mu_2\)) and the alternative hypothesis (\(H_1\)) that there is a difference (\(\mu_1
eq \mu_2\)).
Second, calculate the pooled variance (\(s_p^2\)) using the sample variances and sample sizes, as variances are assumed to be equal. Then, compute the standard error of the difference between the means using the pooled variance.
Third, calculate the z-test statistic using the difference in sample means and the calculated standard error.
Finally, compare this test statistic to the critical z-value from the standard normal distribution at a chosen significance level (\(\alpha\)). For a two-tailed test, the critical values are \(\pm z_{\alpha/2}\). If the absolute value of the test statistic exceeds the critical value, we reject \(H_0\), concluding there is a significant difference between the means.
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