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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?”
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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.
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. Significance level, a
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!
