10.1: Confidence Intervals Falls under the topic of “Inference.” Inference means we are attempting to answer the question, “How good is our answer?” Mathematically: Confidence Interval = Estimate ± M.O.E Conceptually: In repeated sampling, it is the expected percentage of intervals that would “trap” the parameter. Or, “We arrived at this interval using a method that yields correct results 95% of the time.” IT DOES NOT MEAN: Our answer is 95% correct. Our Estimate is our summary statistic, usually the sample mean or sample proportion.

10.1: Confidence Intervals The Margin of Error shows how accurate we believe our guess is, based on the variability of the estimate. Recall from the C.L.T. that the sample mean is Normal with a mean µ and standard deviation Then our Confidence Interval for µ is: z* is determined by the level of confidence we desire. Remember, the data MUST be from an SRS. Outliers can affect the confidence interval. (Why?) The margin of error only covers the natural variation of the data. It DOES NOT help with nonresponse, undercoverage, and other user errors...

Answers the question: “How likely is this sample statistic if we think we know the parameter?” H o : Null Hypothesis - The given parameter value H A : Alternative Hypothesis - Our expectation or hope YOU MUST ALWAYS STATE WHAT YOUR HYPOTHESES ARE. Test Statistic - Standardized value that measures the deviation from our sample statistic to our parameter value. In the case of means: P-value: Probability that if our null hypothesis is true we would get a sample statistic as extreme or more extreme than what we observed. 10.2: Significance Testing

Small P-value means that the odds of getting the sample statistic we did are unlikely given our parameter value. The smaller the P-value, the stronger the evidence AGAINST the null hypothesis. Two possibilities: Either the null hypothesis is wrong or we got an unrepresentative sample. We have to decide in advance how small a probability will allow us to reject the null hypothesis. This is called the “significance level.” We use the Greek letter alpha to represent this. So, if P ≤ alpha, we reject the null hypothesis

10.2: Significance Testing Remember, “statistically significant” means we have evidence to reject the null hypothesis. FOUR STEPS FOR PERFORMING SIGNIFICANCE TEST State your null and alternative hypotheses Calculate the test statistic. (Be sure to show formula) Find the P-value and state a conclusion. EXAMPLES OF PROPER CONCLUSIONS: At the [alpha] level of signficance, we have sufficient evidence to reject the null hypothesis. [P-value ≤ alpha] At the [alpha level of signficance, we do not have sufficient evidence to reject the null hypothesis. [P-value > alpha]

10.2: Significance Testing In Chapter 10 we are often testing the null hypothesis of: H o : µ = µ 0 where µ 0 is the given parameter value So then there are three possibilities for the P-value because there are three place where there can be error. P-value calculation H A : µ > µ 0 ; P-value is P(Z ≥ z) [One sided, right tail] H A : µ < µ 0 ; P-value is P(Z ≤ z) [One sided, left tail] H A : µ ≠ µ 0 ; P-value is 2P(Z ≥ |z|) [Two sided, both tails] Note, for the last one this is the same thing as doing: 2(1- P(Z ≤ |z|))

10.2: Significance Testing EXAMPLE: Do middle-aged male executives have different blood pressure than the general population? Suppose an SRS of 72 executives (aged 35 to 44) is done and the mean is Is this evidence that the executive blood pressures differ from the national average (for males, 35-44) of 128? Test at the 5% level of significance. SOLUTION: Hypotheses: H o : µ = 128H A : µ ≠ 128 Test Statistic: z = P-value: P = 2P(Z ≥ |-1.09|) 2(1 - P(Z ≤ 1.09)) = There is not sufficient evidence that the blood pressure of middle-aged male executives differ from the general population.

10.2: Significance Testing Remember, you can NEVER accept the null hypothesis. We are not showing the null hypothesis to be true. We can only “reject” or “fail to reject.” Also, make sure you always state your conclusion in terms of the question asked. Saying “P-value ≤ alpha” earns zero points Be careful about whether your test is one-sided or two- sided. If you see words like “different” or “changed,” then it’s probably a TWO-SIDED test. If you see words like “higher,” “lower,” “better,” or “worse,” then it’s probably a ONE-SIDED test.