Question

question 5 1 pts which one of the following statements is true? a very small p - value always proves that the null hypothesis is false. if the p - value is small, this means the sample result would occur less than 5% of the time if the null hypothesis is true. the smaller the sample size, the larger the test statistic will be. if the p - value is equal to 0.05, this means there is a 95% chance the alternative hypothesis is true. none of the above statements are true.

Explanation:

• For the first option: A small \(P -\)value provides evidence against the null hypothesis but doesn't "prove" it's false. There's always a chance of Type - I error (rejecting a true null hypothesis).
• For the second option: The \(P -\)value is the probability of getting a sample result (or more extreme) if the null hypothesis is true. A small \(P -\)value (e.g., \(P<0.05\)) means the sample result is unlikely under the null hypothesis, but it's not necessarily less than \(5\%\) (it could be \(P = 0.03\) etc., but the general definition is the probability of the sample result (or more extreme) under \(H_0\), not specifically related to a \(5\%\) threshold in its definition).
• For the third option: The relationship between sample size and test statistic is not straightforward. For example, in a \(z -\)test for a proportion \(z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\), as \(n\) decreases, the denominator \(\sqrt{\frac{p_0(1 - p_0)}{n}}\) increases (assuming \(p_0\) is fixed), and \(z\) (the test statistic) decreases (in magnitude) if \(\hat{p}-p_0\) is fixed.
• For the fourth option: The \(P -\)value is the probability of the data (or more extreme data) given the null hypothesis \(H_0\), not the probability of the alternative hypothesis \(H_1\) being true.

Answer:

None of the above statements are true.