# In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to cons

Significance tests
In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to construct a confidence interval from sample data. Let * be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean: For a two-tailed hypothesis test with level of significance a and null hypothesis H_0 : mu = k we reject Ho whenever k falls outside the c = 1 — alpha confidence interval for mu based on the sample data. When A falls within the c = 1 — alpha confidence interval. we do reject H_0. For a one-tailed hypothesis test with level of significance Ho : mu = k and null hypothesiswe reject Ho whenever A falls outsidethe c = 1 — 2alpha confidence interval for p based on the sample data. When A falls within thec = 1 — 2alpha confidence interval, we do not reject H_0. A corresponding relationship between confidence intervals and two-tailed hypothesis tests is also valid for other parameters, such as p,mu1 — mu_2, and p_1, - p_2. (b) Consider the hypotheses H_0 : p_1 — p_2 = O and H_1 : p_1 — p_2 != Suppose a 98% confidence interval for p_1 — p_2 contains only positive numbers. Should you reject the null hypothesis when alpha = 0.05? Why or why not?

The-ntiligpediestes $$H_{0}: p_{1}-p_{2}=90$$
$$H_{0}: p_{1}-p_{2}>0$$
Here, from above hypothesis $$p_{1} — p_{2} = 0$$ and we know that for a one-tailed hypothesis test with level of significance a, we reject $$H_{0}$$ whenever the difference of proportions falls outside the $$c= 1-\alpha$$ confidence interval for p based on the sample data. If a 98% confidence interval for $$p_{1} — p_{2}$$ contains only positive numbers then we should reject $$H_{0}\ at\ \alpha = 0.02$$ because the confidence interval does not contain 0. We know that 99% confidence interval is greater than 98% confidence interval and 99% confidence intervalmight contain 0. So, we don’t have enough evidence to reject null hypothesis $$H_{0}\ at\ \alpha = 0.01$$ level of significance.