How do you calculate and interpret a p-value and a confidence interval in hypothesis testing?

How do you calculate and interpret a p-value and a confidence interval in hypothesis testing?

• A. A p-value indicates the probability of observing the data if the null hypothesis is true; a confidence interval provides a range of plausible values for the parameter; if the p-value is below alpha, the result is statistically significant, and the CI excludes the null value.
• B. P-value is the effect size; confidence interval is the sample size.
• C. P-value is the probability the null is true; CI is the p-value range.
• D. If p-value below alpha, reject the alternative hypothesis.

Answer: (A)

The main idea is how data provide evidence about a hypothesis and about a parameter through two complementary tools: the p-value and the confidence interval. The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic as extreme or more extreme than what was observed. It tells you how compatible your data are with the null. A small p-value means the observed result would be unlikely if the null were true, so you have evidence against the null at your chosen significance level. It is not the probability that the null is true, and it does not measure the size of an effect. A confidence interval, on the other hand, gives a range of parameter values that are consistent with the data at a given confidence level (like 95%). If you repeated the study many times, 95% of those intervals would capture the true parameter. The interval conveys both an estimate (the center) and the precision (the width). A key connection is that, for the usual tests, if the p-value is below your alpha threshold, the null value lies outside the corresponding confidence interval. That coherence helps you see how a small p-value aligns with a CI that does not include the null value. An example helps: testing whether a mean equals a specific value, you compute a test statistic and a p-value from assuming that mean, and you also compute a 95% CI for the mean. If the null value is not in that 95% CI, the p-value would be less than 0.05, indicating statistical significance at that level. Why the other ideas don’t fit: the p-value is not an effect size, and the confidence interval is not a statement about sample size. Saying the p-value is the probability the null is true or that you reject the alternative reverses the actual logic of hypothesis testing.

The main idea is how data provide evidence about a hypothesis and about a parameter through two complementary tools: the p-value and the confidence interval.

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic as extreme or more extreme than what was observed. It tells you how compatible your data are with the null. A small p-value means the observed result would be unlikely if the null were true, so you have evidence against the null at your chosen significance level. It is not the probability that the null is true, and it does not measure the size of an effect.

A confidence interval, on the other hand, gives a range of parameter values that are consistent with the data at a given confidence level (like 95%). If you repeated the study many times, 95% of those intervals would capture the true parameter. The interval conveys both an estimate (the center) and the precision (the width).

A key connection is that, for the usual tests, if the p-value is below your alpha threshold, the null value lies outside the corresponding confidence interval. That coherence helps you see how a small p-value aligns with a CI that does not include the null value.

An example helps: testing whether a mean equals a specific value, you compute a test statistic and a p-value from assuming that mean, and you also compute a 95% CI for the mean. If the null value is not in that 95% CI, the p-value would be less than 0.05, indicating statistical significance at that level.

Why the other ideas don’t fit: the p-value is not an effect size, and the confidence interval is not a statement about sample size. Saying the p-value is the probability the null is true or that you reject the alternative reverses the actual logic of hypothesis testing.