Chapter 11 Testing a Claim. AP Statistics Hamilton/Mann. Introduction. Confidence intervals are one of the two most common types of statistical inference. Use confidence intervals when you want to estimate a population parameter.
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Significance Tests: The Basics
HW: 11.3, 11.4, 11.5, 11.6, 11.7, 11.8, 11.11, 11.12, 11.14, 11.16…due Tuesday at the end of class
We can judge whether any observed value is unusual by locating it on this distribution.
We abbreviate the null hypothesis as H0 and the alternative hypothesis as Ha. In our example, we were seeking evidence of a decrease in response time this year. For this reason, the null hypothesis would say that there was “no decrease” and the alternative hypothesis would say that “there was a
decrease.” The hypotheses are
Significant in the statistical sense does not mean important. It simply means “not likely to happen by chance.” The significance level α makes “not likely” more precise. Significance at level 0.01 is often expressed by the statement “The results were significant (P < 0.01).” Here the P stands for the P-value. The P-value is more informative than a statement of significance because it allows us to assess significance at any level we choose. For example, a result with P=0.03 is significant at the α = 0.05 level, but is not significant at the α = 0.01 level.
Carrying Out Significance Tests
HW: 11.27, 11.28, 11.30, 11.31, 11.32, 11.33, 11.34
Once you have completed the first two steps, a calculator or computer can do Step 3. Here is the calculation step for carrying out a significance test about the population mean μ in the unrealistic setting when σ is known.
The link between confidence intervals and two-sided significance tests is sometimes referred to as duality.
Use and Abuse of Tests
HW: 11.43, 11.44, 11.45, 11.46, 11.48
Using Inference to Make Decisions
HW: 11.49, 11.50, 11.51, 11.52, 11.53, 11.54, 11.55, 11.56, 11.57
If H0 is true, our decision is either correct (if we fail to reject H0) or is a Type I error.
The shaded area represents the probability of failing to reject H0 when μ = 6.4 minutes. So this area is the probability of a Type II error.
The probability of a Type II error (sometime called β) for the particular alternative μ = 6.4 minutes is the probability the test will fail to reject H0 when μ = 6.4 minutes. This is the probability that the sample mean falls to the right of the critical value.