Hypothesis Testing

Another inference method We’ve used confidence intervals to give an estimate (with a margin of error) of m. We change the question we’re asking… from, “What’s an interval that likely encloses the parameter?” to, “Is the parameter equal to a certain value, or in some way different?”

Null hypothesis Null hypothesis is always of the form“parameter = #” e.g., m = 20 oz. Also called H0 (read: “H naught”) We need evidence to make us reject this hypothesis. H0 is formulated prior to collecting data.

Alternative hypothesis Takes 1 of 3 forms “parameter  #” (two-sided) “parameter > #” (one-sided) “parameter < #” (one-sided) e.g., m < 20 oz. Also called Ha Must acquire evidence in favor of Ha before rejecting H0. Ha is formulated prior to collecting data.

Test statistic Calculated based on sample data and on H0. How far is what you observed away from what you would expect if H0 were true? Uses information about the mean and standard deviation of the sampling distribution of your estimator ( , for example).

Example: Fabric Strength A vendor submits lots of fabric to a textile manufacturer. If the average breaking strength of a lot exceeds 200 psi, the manufacturer will accept the lot. Past experience indicates that the standard deviation of breaking strength is 10 psi. 20 specimens are randomly chosen; the average breaking strength of these is 204 psi. Define null and alternative hypotheses for this setting. Compute a test statistic for this situation. What assumption(s) do you need to make?

Calculating p-values Assume H0 is true. Now, calculate the probability of seeing something as extreme as what you observed or more extreme. “Extreme” depends on Ha. Use information about the sampling distribution of the estimator!

Ha: m > # Ha: m < # Ha: m  #

Interpreting p-values The p-value is the probability of observing something as extreme as your data (or more so) under H0. The smaller the p-value, the less credibility you give to H0 (more to Ha). If the p-value is large, then your observed data is close to what you would expect if H0 were true.

Significance level, a We need to compare the p-value to a fixed value, a (chosen in advance). a is related to the amount of evidence we will require to reject H0. The closer a is to zero, the more evidence we require to reject H0. a is the probability of falsely rejecting H0.

Assessing statistical significance If p-value < a, we reject H0. We say that the data are statistically significant at significance level a. The p-value is the smallest level a at which the data are significant. It’s more informative than the final decision: “reject H0” or “fail to reject H0”.

Cautions about hypothesis testing Choose hypotheses and level of significance carefully, prior to collecting data. Don’t ignore lack of significance, particularly if p-value is close to a. Even if we have a significant result, the difference from H0 may be very small. If experiment/survey is poorly designed, hypothesis testing won’t help!