Which of the following statements is correct?
A. An extremely small p-value indicates that the actual data differs markedly from that expected if the null hypothesis were true.
B. A large p-value indicates that the data is consistent with the alternative hypothesis.
C. The p-value measures the probability of making a Type II error.
D.The larger the p-value, the stronger the evidence against the null hypothesis
E. The p-value measures the probability that the hypothesis is true.
The correct statement among the options given is:
A. An extremely small p-value indicates that the actual data differs markedly from that expected if the null hypothesis were true.
Explanation:
A p-value is a statistical measure used in hypothesis testing. It measures the strength of evidence against the null hypothesis. In hypothesis testing, the null hypothesis assumes that there is no significant difference between observed data and expected data, whereas the alternative hypothesis states that there is a significant difference.
A small p-value (usually less than the predetermined significance level, commonly 0.05) indicates that the observed data differs significantly from what is expected under the null hypothesis. Therefore, it provides evidence to reject the null hypothesis in favor of the alternative hypothesis.
The correct statement among the options provided is A. An extremely small p-value indicates that the actual data differs markedly from that expected if the null hypothesis were true.
To understand why this statement is correct, it's important to understand what a p-value represents. In statistical hypothesis testing, the p-value is a measure of the strength of evidence against the null hypothesis. The null hypothesis assumes that there is no difference or relationship between variables, while the alternative hypothesis assumes that there is a difference or relationship.
A p-value tells us the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming that the null hypothesis is true. If the p-value is extremely small (smaller than the predetermined significance level, often 0.05), it suggests that the observed data is unlikely under the null hypothesis. Therefore, we reject the null hypothesis in favor of the alternative hypothesis because there is strong evidence that the actual data differs markedly from what we would expect if the null hypothesis were true.
Let's look at the other options to understand why they are not correct:
B. A large p-value indicates that the data is consistent with the alternative hypothesis.
This statement is incorrect. A large p-value actually suggests that the observed data is likely under the null hypothesis, meaning that there is weak evidence against the null hypothesis. It does not support the alternative hypothesis.
C. The p-value measures the probability of making a Type II error.
This statement is incorrect. The p-value represents the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. It is not directly related to Type II error, which is the probability of failing to reject the null hypothesis when it is false.
D. The larger the p-value, the stronger the evidence against the null hypothesis.
This statement is incorrect. The opposite is true. A small p-value provides stronger evidence against the null hypothesis, as it indicates that the observed data is unlikely to occur if the null hypothesis were true.
E. The p-value measures the probability that the hypothesis is true.
This statement is incorrect. The p-value is not a measure of the probability that a hypothesis is true. It only provides information about the strength of evidence against the null hypothesis.