Question

1. which one of the following statements is true?

a. a very small p - value always proves that the null hypothesis is false.
b. 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.
c. the smaller the sample size, the larger the test statistic will be.
d. if the p - value is equal to 0.05, this means there is a 95% chance the alternative hypothesis is true.
e. none of the above statements are true.

Explanation:

• Option A: A small \(P -\)value provides evidence against the null hypothesis but does not "prove" it is false. There is always a possibility of Type - I error (rejecting a true null hypothesis).
• Option B: The \(P -\)value is the probability of obtaining a sample result as extreme or more extreme than the one observed, assuming the null hypothesis is true. A small \(P -\)value (e.g., \(P<0.05\)) means that the sample result is unlikely to occur if the null hypothesis is true. The \(5\%\) is an arbitrary common significance level, but the \(P -\)value itself is the probability of the sample result under the null hypothesis.
• Option C: 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 \(n\) (sample size) decreases, the standard error \(\sqrt{\frac{p_0(1 - p_0)}{n}}\) increases. If \(\hat{p}-p_0\) is positive, \(z\) (test statistic) decreases as \(n\) decreases.
• Option D: The \(P -\)value is not the probability that the alternative hypothesis is true. It is a probability statement about the data under the null hypothesis.

Answer:

B. 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.