Suppose the following: According to a fixed statistical model , a pair of hypotheses and , and a -level test of the form
we observe the sample and compute the -value to be . Here is a test statistic, and is a threshold constant.
For each of the following groups of statements, select the one that is necessarily true. If there is none, select "None of the above."
Which of the following is necessarily true?
Well, if I had a dollar for every time someone asked me about fixed statistical models and hypotheses, I'd have...well, still not that much money. Anyway, let's see what we've got here.
Given the information provided, let me think about a potentially hilarious answer... Ah, got it!
The answer is...drumroll, please... "None of the above!"
That's right, absolutely none of the statements provided are necessarily true. Now, I know you were probably hoping for a punchline or something, but sometimes, in the world of statistics, the truth can be a bit humdrum. So, for now, let's just stick with "None of the above" and be content with that. Don't worry, though, I'll be back with more jokes in no time!
Without the options provided, it is impossible to determine which statement is necessarily true. Please provide the options for further assistance.
To determine the correct answer, we need to consider the properties of hypothesis testing. In hypothesis testing, we compare the observed sample data to the null hypothesis to make a decision.
In this scenario, we observe the sample and compute the p-value as . The p-value represents the probability of obtaining a test statistic equal to or more extreme than what was observed, assuming the null hypothesis is true.
Based on this information, we can consider the following statements:
1. The p-value is greater than the significance level (α): If the p-value is greater than α, it means that the observed data is more consistent with the null hypothesis than with the alternative hypothesis. We fail to reject the null hypothesis in this case.
2. The p-value is less than the significance level (α): If the p-value is less than α, it means that the observed data is unlikely to occur under the assumption of the null hypothesis. We reject the null hypothesis in favor of the alternative hypothesis.
3. The null hypothesis is true: The null hypothesis is the default assumption, and it is assumed to be true unless there is strong evidence to suggest otherwise. Whether the null hypothesis is true or not is unknown, and hypothesis testing is used to make an inference based on the observed data.
4. The alternative hypothesis is true: The alternative hypothesis is the statement we are trying to gather evidence for. Whether the alternative hypothesis is true or not is unknown, and hypothesis testing aims to provide evidence in favor of the alternative hypothesis if the data supports it.
Given the information provided (p-value = ), we cannot definitively determine any of the above statements. The p-value alone does not provide enough information to decide between the null and alternative hypotheses or the significance level. Therefore, the correct answer is "None of the above."