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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
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asked 2020-12-24

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 \(c = 1 — 2\alpha\) confidence interval for p based on the sample data. When A falls within the \(c = 1 — 2\alpha\) 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 \(p,\mu_1 — \mu_2,\) 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?

Expert Answers (1)

2020-12-25
The-ntiligpediestes \(H_{0}: p_{1}-p_{2}=90\)
The alternative hypothesis:
\(H_{0}: p_{1}-p_{2}>0\)
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.
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