If a report states that certain data were used to reject a given hypothesis, would it be a good idea to know what type of test (one-tailed or two-tailed) was used? Explain.

Trent Carpenter 2021-03-05 Answered
If a report states that certain data were used to reject a given hypothesis, would it be a good idea to know what type of test (one-tailed or two-tailed) was used? Explain.

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Expert Answer

Bentley Leach
Answered 2021-03-06 Author has 17093 answers
A report states that certain data were used to reject a given hypothesis, it would be a good idea to know what type of test (one-tailed or two- tailed) was used. The conclusions can be different for two tailed and one tailed test.
The conclusion based on the two-tailed test is more conservative.
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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 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?

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When σ is unknown and the sample size is \(\displaystyle{n}\geq{30}\), there are tow methods for computing confidence intervals for μμ. Method 1: Use the Student's t distribution with d.f. = n - 1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When \(\displaystyle{n}\geq{30}\), use the sample standard deviation s as an estimate for σσ, and then use the standard normal distribution. This method is based on the fact that for large samples, s is a fairly good approximation for σσ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution. Consider a random sample of size n = 31, with sample mean x¯=45.2 and sample standard deviation s = 5.3. (c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?
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