STATS 250 1st Edition Lecture 9 Outline of Last Lecture I. CI Module 1: Confidence Interval for a Population Proportion p, part 2II. Using Confidence Intervals to Guide DecisionsOutline of Current Lecture I. HT Module 0: an Overview of Hypothesis TestingII. HT Module 1: Testing Hypotheses About a Population ProportionIII. If n is SmallCurrent LectureI. HT Module 0: an Overview of Hypothesis Testinga. Null Hypothesis, H0: a statement that there is nothing happeningi. In most situations, the researcher hopes to disprove or reject the null hypothesisii. A generic null hypothesis could be H0: population parameter = null value, where the null value is the specific number the parameter equals if the null hypothesis is trueiii. We assume the null hypothesis is true until the sample data conclusively demonstrates otherwiseb. Alternative Hypothesis, Ha: statement that something is happeningc. Direction of Extreme: indicated by the alternative hypothesis, shows that there is a skewed directioni. One-Sided/One-Tailed Hypothesis Test: when the alternative hypothesis specifies a single direction ii. Two-Sided/Two-Tailed Hypothesis Test: the alternative hypothesis includes values in both directions from a specific standardd. Data is summarized via a test statistic, which is often a standardized statistic measuring the distance between the sample statistic and the null value in standard error unitsi. Test Statistic = Sample Statistic−Null Value(Null)Standard Errorii. p-value: computed by assuming the null hypothesis is true and then determining the probability of a result as extreme (or more extreme) as the observed test statistic in the direction of the alternative hypothesis1. Is a probability, so it must be between 0 and 12. Is not that probability that the null hypothesis is true!3. If the p value is ≤ a, we reject H0, and say the results are statistically significant at the level a4. If p-value is > a, we fail to reject H0, and say the results are not statistically significant at the level aII. HT Module 1: Testing Hypotheses About a Population Proportiona. Step 1: Determine the null and alternative hypothesesb. Step 2: Verify necessary data conditions, and if met, summarize the data into an appropriate test statistici. The data are assumed to be in a random sampleii. Check if np0 ≥ 10 and n(1- p0) ≥10iii. Observed test statistic: z = ^p− p0√p0(1− p0)nc. Step 3: Assuming the null hypothesis is true, find the p-valued. Step 4: Decide if the result is statistically significant based on the p-valuee. Step 5: Report the conclusion in the context of the situationIII. If n is Smalla. If sample size is small, we go back to the exact distribution for a count X, called the binomial distributionb. Use the binomial probability formula for computing exact p-valuei. X = the number of successes in the sample size n which has the Bin(n,p0) distribution when H0 is