Question

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Using your statistics class as a sample to see if there is evidence of a difference between male and female students in how many hours are spent watching television per week.

Short answer

Considering the context of this exercise, which doesn't seem to carry high stakes, using a relatively larger significance level such as alpha = 0.10 may be appropriate as it minimizes the risk of Type II error, and gives a better chance of detecting a difference if one truly exists.

Step-by-step solution

Understanding the Null and Alternate Hypotheses

In any statistical study, the first step is often to formulate the null and alternate hypotheses. In this case, the null hypothesis can be that there is no difference in the average television watching time between male and female students. The alternative hypothesis is the opposite of the null hypothesis and in this case, it would be that there is a significant difference between the averages.

Assessing the Potential Risks of Type I and Type II errors

The choice of significance level (alpha) is linked to the risks of type I and type II errors. As a researcher, one needs to understand the implications of committing either of these errors. In the context of this exercise, a Type I error (false positives) would incorrectly suggest that there is a difference in the watching habits of male and female students when in reality there isn't. In contrast, a Type II error (false negatives) would incorrectly suggest that there are no differences between the two groups when in fact there are.

Choose the Appropriate Significance Level

Now, considering the potential effects of type I and II errors, one can decide on the most appropriate significance level. If claiming a difference where none exists (false positives, Type I) would significantly impact the subsequent actions or decisions, a smaller alpha, say 0.01, should be used. If failing to detect a difference where one exists (false negatives, Type II) would carry more consequences, a larger alpha of 0.10 might be chosen. As this statistic belongs to a classroom research and likely doesn't carry high stakes, a larger α=0.10 could be used as it would minimize the probability of a Type II error, giving a better chance of detecting a difference if it truly exists.

Key concepts

Null and Alternate Hypotheses

Understanding the null and alternate hypotheses is essential in any statistical analysis, as it lays the foundation for testing a theory. The null hypothesis (\(H_0\)) is a statement of no effect or no difference, typically representing the status quo or a position of skepticism. For example, if we were testing if a drug has an effect on a health condition, the null hypothesis would state that the drug has no effect.

Conversely, the alternate hypothesis (\(H_1\) or \(H_a\)), proposes that there is an effect or a difference. Continuing with our example, the alternate hypothesis would contend that the drug does improve the health condition. When conducting a test of hypothesis, statistical evidence is used to determine whether to reject the null hypothesis in favor of the alternate hypothesis. The outcome hinges on the level of significance and the data collected. In educational settings, such as comparing study hours between genders as in our exercise, these hypotheses guide the investigation and conclusions drawn about student behaviors.

Type I and Type II Errors

In the realm of statistics, not all errors are made equal. There are two specific kinds of errors to consider when testing hypotheses: Type I and Type II errors. A Type I error occurs when the null hypothesis is true, but is incorrectly rejected. It’s akin to a false positive in medical tests, where someone is told they have a condition when they do not. The significance level (\( \(\alpha\) \) ), directly controls the probability of a Type I error; a lower significance level means a lower chance of making this error.

In contrast, a Type II error, symbolized by \(\beta\), happens when the null hypothesis is false, but we fail to reject it. This is similar to a false negative, where a real effect is missed. The power of the test, which is 1 - \(\beta\), determines the test's ability to detect an effect when there is one. Consequently, scientists and researchers must carefully balance the risks of these errors when deciding the appropriate significance level, often prioritizing which error is more critical to avoid in the context of their study.

Statistical Significance

Statistical significance is a determination of whether the observed effect or difference in a test is unlikely to have occurred by chance. It is an indicator of how confident we can be in the results of our statistical analysis. A result is said to be statistically significant if it is unlikely to have happened under the null hypothesis, typically assessed by a p-value. The p-value represents the probability of observing the data, or something more extreme, assuming that the null hypothesis is true.

If the p-value is less than or equal to the chosen significance level (\( \(\alpha\) \) ), we reject the null hypothesis, suggesting that there is statistically significant evidence against it. Selecting an \(\alpha\) level is subjective, but common values are 0.05, 0.01, or 0.10, as mentioned in the exercise. It depends on the context of the study and how serious a Type I error would be. Research with significant consequences often uses a more stringent significance level, such as 0.01, to minimize the risk of false positives. In contrast, exploratory or preliminary research, where the aim is to detect possible signals that warrant further investigation, may use a higher threshold like 0.10.