Chapter 9 – Statistical Inference: Significance Tests about Hypotheses (Hypothesis Tests)Confidence Intervals are useful if we don’t have a question in mind or an idea about what the true value of the population parameter may be. If you do have a statistical question about the nature of the population parameter, however, you can conduct a significance test to check yourself.No matter what the parameter you seek to investigate, you will always follow the same five stepswhen conducting a hypothesis test. Only the specifics within the steps will change.Definition: a hypothesis is a statement about the population distribution parameter(s), usually claiming that it takes on a specific value or falls in a range of values.The steps of a HT (these differ slightly from the book).1. [H] State the hypotheses, being sure to define the parameter in context. There are two hypotheses to include. The null hypothesis (H0) states the parameter takes on a particular value, and represents no (null) effect. The alternative hypothesis (HA) states the parametercan be any of the remaining possible values for the parameter, that some effect takes place.2. [T] Identify the test appropriate for testing the hypothesis.3. [A] State (and check) your assumptions for the test. Once the hypotheses have been established there are assumptions that accompany the appropriate test.4. [M] Mechanicsa. state the significance level (α = 0.05 unless specified by the problem)b. identify the rejection rule (reject H0 if p-value < α)c. calculate the test statistic d. calculate the p-value5. [C] State the conclusion for the test, including the decision, justification, and contextDefinition: the p-value is a probability summarizing the chance of obtaining a point estimate at least as extreme as the observed value, assuming the parameter value specified in the null hypothesis is true (and where extreme is defined by the alternative hypothesis). This is based on the sampling distribution of the point estimate.Smaller p-values are evidence against H0.Hypothesis Test for Population Proportion (p)If one is interested in testing for a proportion, the steps are:[H] H0: p = p0HA: p ≠ p0 or (p < p0) or (p > p0)[T] One-sample Z-test for population proportion[A] - categorical observations- simple random sample- np0 and n(1-p0) are both at least 15[M]a. α = 0.05b. reject H0 if p-value < α = 0.05c.zobs=(^p−p0)/√p0(1− p0)/nd. p-value = 2*Pr[Z < -|zobs|] if two-tailed (p ≠ p0)p-value = Pr[Z < zobs] if left-tailed (p < p0)p-value = Pr[Z > zobs] if right-tailed (p > p0)[C] Reject/do not reject the null hypothesis that the true proportion of success is p0 because the p-value is/is not less than α = 0.05.Example: You are suspicious, based on anecdotal experience, of Wrigley’s claim of 20% red Skittles. You collect a random sample of 100 Skittles that has 15 red Skittles. Test the company’s claim at the 0.05 level.[H] H0: p = .2HA: p ≠ .2where p is the true proportion of red Skittles candies[T] One-sample Z-test for population proportion[A] - categorical observations- simple random sample- np0 and n(1-p0) are both at least 15[M]a. α = 0.05 (given in problem)b. reject H0 if p-value < α = 0.05c.zobs=^p− p0√p0(1−p0)n=.15−.2√.2(.8)100=−.05.04=−1.25d. p-value = 2*Pr[Z < -|-1.25|] = 2*Pr[Z < -1.25] = 2*0.1056 = 0.2112[C] Do not reject the null hypothesis that the true proportion of red Skittles is 0.20 because the p-value = 0.2112 is not less than α = 0.05. There is insufficient evidence to reject H0.Hypothesis Test for Population Mean (µ)If we are measuring quantitative variables, we follow the same basic steps for inference about thepopulation mean. Only a few details change:[H] H0: µ = µ 0HA: µ ≠ µ 0 or (µ < µ 0) or (µ > µ 0)[T] One-sample T-test for population mean (Note: we are again using a T distribution for the sampling distribution because it is unlikely that we will know the population standard deviation)[A] - quantitative observations- simple random sample- approximately normal population distribution[M]a. α = 0.05b. reject H0 if p-value < α = 0.05c.tobs=´x−μ0s√n, which has υ = n – 1 degrees of freedomd. p-value = 2*Pr[Tυ < -|tobs|] if two-tailed (µ ≠ µ 0)p-value = Pr[Tυ < tobs] if left-tailed (µ < µ 0)p-value = Pr[Tυ > tobs] if right-tailed (µ > µ 0)[C] Reject/do not reject the null hypothesis that the true population mean is µ0 because the p-value is/is not less than α = 0.05.Example:A randomly selected team of 16 female huskies has a mean weight of 44 lbs and standard deviation of 1lb. Test the hypothesis that female huskies weigh at most 42lbs on average.[H] H0: µ ≤ 42lbsHA: µ > 42lbswhere µ is the true average weight of female huskies, in pounds.[T] One-sample T-test for population mean (Note: we are again using a T distribution for the sampling distribution because it is unlikely that we will know the population standard deviation)[A] - quantitative observations- simple random sample- approximately normal population distribution[M]a. α = 0.05 (by default because not provided)b. reject H0 if p-value < α = 0.05c.tobs=44−421√16=214=8 and υ = n-1 = 16-1 = 15 degrees of freedomd. p-value = Pr[Tυ > tobs] = Pr[T15 > 8] ´¿0[C] Reject the null hypothesis that the true mean weight of female huskies is at most 42lbs because the p-value is less than α = 0.05. There is sufficient evidence in favor of the alternative hypothesis to reject H0.Statistical vs. Practical SignificanceJust because something comes up as statistically significant according to a hypothesis test, it maynot have real-world meaning or application.Thought exercise:Kramer normally drives a golf ball 190 yards on average, and is pretty consistent (σ = 5yd). Kramer has 600 Titleist golf balls that he takes to the Rockaways, and averages 191 yards on those drives.This is a statistically significant result, but is it practically significant? Maybe, if he’s trying to break into the pro circuit. But the results were largely driven by the large sample size and not by any meaningful change in the golf balls’ average driving distance.Hypothesis Tests and Types of ErrorsOf course, whenever we make a decision, we could be wrong. And, we’re wrong with some probability. We can quantify the chance of the types of errors we can make, but first we have to understand those errors.State of the Real WorldH0 True H0 FalseTestDecisionH0 True(Fail to Reject)Correct DecisionType II Error(β)H0