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Chapter 4: Problem 160

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1} \neq \mu_{2} .\) In addition, in each case for which the results are significant, state which group ( 1 or 2 ) has the larger mean. (a) \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : 0.12 to 0.54 (b) \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -2.1 to 5.4 (c) \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -10.8 to -3.7

Short Answer

Expert verified
Based on the given confidence intervals, we reject the null hypothesis for samples (a) and (c) and fail to reject it for sample (b). In case (a), group 1 has a larger mean. In case (b), we cannot determine which group has a larger mean. In case (c), group 2 has a larger mean.

Step by step solution

01

Interpret Confidence Interval (a)

A 95% confidence interval for \( \mu_{1} - \mu_{2} \) ranges from 0.12 to 0.54. This interval excludes 0, hence we reject the null hypothesis \( H_0: \mu_{1} = \mu_{2} \) at the 5% significance level. The positive difference suggests that group 1 has a larger mean.
02

Interpret Confidence Interval (b)

A 99% confidence interval for \( \mu_{1} - \mu_{2} \) ranges from -2.1 to 5.4. This interval contains 0, hence we fail to reject the null hypothesis \( H_0: \mu_{1} = \mu_{2} \) at the 1% significance level. We cannot conclude which group has a larger mean.
03

Interpret Confidence Interval (c)

A 90% confidence interval for \( \mu_{1} - \mu_{2} \) ranges from -10.8 to -3.7. This interval also excludes 0; hence we reject the null hypothesis \( H_0: \mu_{1} = \mu_{2} \) at the 10% significance level. The negative difference suggests that group 2 has a larger mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval Interpretation
When you're interpreting a confidence interval, you're essentially trying to understand the range within which you can expect your statistical parameter to fall.
  • A confidence interval provides an estimated range derived from a sample dataset.
  • It shows where you might find the true difference between the means in the whole population if you sampled the population over and over again.
For instance, if a 95% confidence interval for \( \mu_1 - \mu_2 \) ranges from 0.12 to 0.54, it means that we are 95% confident that the true mean difference is somewhere between these numbers. Crucially, if a confidence interval doesn't include 0,it suggests that there is a significant difference between the means of the two groups. In other words, the null hypothesis can often be rejected, indicating that a real effect or difference exists between the groups. Conversely, if 0 is inside the interval, it implies there's no significant difference, as seen in example (b) where the interval was -2.1 to 5.4.Here, we conclude that we cannot reject the null hypothesis.
Significance Level
The significance level, denoted by \( \alpha \), is a threshold set by the researcher which determines how strong the evidence must be to reject the null hypothesis. It's foundational in hypothesis testing because:
  • It reflects our tolerance for making a Type I error: erroneously rejecting a true null hypothesis.
  • A lower significance level implies stricter criteria for rejecting the null hypothesis.
  • Common levels of significance are 0.05, 0.01, and 0.10, corresponding to confidence levels of 95%, 99%, and 90% respectively.
In this context, if our 95% confidence interval for \( \mu_1 - \mu_2 \) is between 0.12 and 0.54,we reject the null hypothesis at a significance level of 5% (\( \alpha = 0.05 \)).It implies that there is less than a 5% probability that the results observed are due to random chance.Significance levels critically aid us in determining whether the observed effects are "real" or merely the consequence of random sampling error.
Null and Alternative Hypotheses
To conduct a statistical test, one starts by defining the null and alternative hypotheses. These form the cornerstone of any hypothesis testing framework.
  • The **null hypothesis** (\( H_0 \): \( \mu_1 = \mu_2 \)) posits that there is no difference between the population means.
  • The **alternative hypothesis** (\( H_a \): \( \mu_1 eq \mu_2 \)) suggests there is some difference.
The null hypothesis serves as the default or status quo, while the alternative hypothesis is what the researcher aims to support.In practice, we use data to test whether we can confidently refute the null hypothesis in favor of the alternative. When the confidence interval for the difference \( \mu_1 - \mu_2 \) excludes zero, we have reason to reject the null hypothesis and thus infer a difference.When it includes zero, however, as in interval case (b) from -2.1 to 5.4, we lack the statistical evidence to reject the null hypothesis, indicating that any observed difference could be due to chance rather than an actual effect.

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Most popular questions from this chapter

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