Question

two independent samples have been selected, observations from population 1 and observations from population 2. the sample means have been calculated to be and . from previous experience with these populations, it is known that the variances are and . (a) find . answer: (b) determine the rejection region for the test of and use . (c) compute the test statistic. the final conclustion is a. we can reject the null hypothesis that and accept that . b. there is not sufficient evidence to reject the null hypothesis that . (d) construct a % confidence interval for .

Answer 1

In statistics, when the population standard deviations are known, a hypothesis test comparing two population means determines if there is a statistically significant difference between them. The null hypothesis states that the means are equal, and it is rejected if the test statistic results in a p-value less than the chosen significance level, such as 5%.

Step-by-step explanation:

When conducting a hypothesis test for two independent samples where the population standard deviations are known, we compare the means from each population (μ1 and μ2). This scenario fits into the subject of statistics within Mathematics at the college level. The hypothesis test will assess whether any observed difference in sample means is statistically significant.

In these cases, the null hypothesis (Φ0) usually states that the two population means are equal (μ1=μ2), and the alternative hypothesis (Φ1) is that they are not equal (μ1≠μ2). The test statistic is calculated using the sample means, the known population standard deviations, and sample sizes.

Based on this test statistic and the chosen significance level (α), we determine whether to reject or fail to reject the null hypothesis. For a 5 percent significance level, if the calculated p-value is less than 0.05, we reject the null hypothesis, indicating there is a statistically significant difference between the two population means.