Critical Value vs P-Value
Both terms below come up in the same part of the course, and students mix them up. Here is each one defined on its own, side by side, so you can see where they part company.
Critical value
Confidence intervals
A critical value is a cutoff from a reference distribution, such as z or t, that sets a confidence interval's width or a test's rejection boundary.
A critical value marks how far out on a distribution you go to capture a chosen probability. For a confidence interval it is the number of standard errors that brackets the middle C% of the sampling distribution. For example, a 95% confidence interval for a mean using the normal model uses , because 95% of the standard normal curve lies within 1.96 standard deviations of the center. With small samples and unknown population spread you instead read from the -distribution using the degrees of freedom.
P-value
Hypothesis testing
A p-value is the probability, computed assuming the null hypothesis is true, of getting a result at least as extreme as the one you observed.
A p-value measures how well your data agree with the null hypothesis: a small p-value means the observed result would be surprising if the null were true, which counts as evidence against it. It is a probability about the data, not the probability that the null hypothesis is true. For example, a p-value of 0.02 says that if the null hypothesis held, data at least this extreme would occur about 2% of the time. You reject the null hypothesis when the p-value is at or below the significance level (the Greek letter alpha).