Question

in order to compare the means of two populations, independent random samples of observations are selected from each population, with the following results: sample 1sample 2 (a) use a % confidence interval to estimate the difference between the population means . -31.78762353 35.78762353 (b) test the null hypothesis: versus the alternative hypothesis: . using , give the following: (i) the test statistic 0.1166584562 (ii) the positive critical score 1.970806 (iii) the negative critical score -1.970806 the final conclusion is we can reject the null hypothesis in favor ofthe alternative . there is not sufficient evidence to reject the null hypothesis . (c) test the null hypothesis: versus the alternative hypothesis: . using , give the following: (i) the test statistic -1.516559931 (ii) the positive critical score 1.970806 (iii) the negative critical score -1.970806 the final conclusion is we can reject the null hypothesis in favor of the alternative . there is not sufficient evidence to reject the null hypothesis .

Answer 1

The difference between sample means is represented by the random variable X₁ - X₂. The null hypothesis, stating no difference between population means, is not rejected if the test statistic is within the critical value range.

Step-by-step explanation:

In comparing the means of two populations, the random variable X₁ - X₂ represents the difference between sample means from each population. When calculating the confidence interval for the difference between the population means, we use the given confidence level to estimate the interval that contains the true difference with a certain degree of certainty.

When performing a hypothesis test, the null hypothesis (H0) typically states there is no difference between the population means, while the alternative hypothesis (H1 or Ha) suggests that there is a difference.

To decide whether to reject the null hypothesis, we compare the test statistic to critical values. If the test statistic falls outside the range defined by the critical values, we reject the null hypothesis. The critical values are based on the chosen significance level (alpha, α).

Concerning the provided example:
(b) Since the test statistic 0.1166584562 is within the critical scores 1.970806 and -1.970806, we conclude that there is not sufficient evidence to reject the null hypothesis.
(c) If the test statistic -1.516559931 is compared against the same critical values at α = 0.05, we similarly conclude that there is not sufficient evidence to reject the null hypothesis. If α were 0.01, the decision might differ as the critical values would be more extreme, typically leading to a higher threshold for rejecting the null hypothesis.