Aem 201 1st Edition Lecture 16PREVIOUS LECTUREI. Interval Estimation of the Population Mean-Population Standard Deviation UnknownII. Interval Estimation of the Population ProportionCURRENT LECTUREI. One-Sampling Hypothesis TestII. Steps in Hypothesis TestingONE-SAMPLING HYPOTHESIS TEST- Hypothesis Testing: scientific method-based means for using sample data to evaluate conjectures about a population- Null Hypothesis: Statement of the conjectured value(s) for the parameter thatincludes (but is not necessarily limited to) equality between the conjectured value and tested parameter usually denoted as:- This is equivalent to a claim that the difference between the observations andhypothesized value are due to random variationo The null hypothesis can only be less than or equal to, equal to or greater than or equal to- Alternative Hypothesis: Statement of the conjectured value(s) for the parameter that is mutually exclusive and collectively exhaustive with respect to the Null Hypothesis and so includes the < and/or > relationship between conjectured value and the tested parameter. Usually denoted as:- This equivalent to the claim that the difference between the observations andthe hypothesized value are systematic (i.e., due to something other than random variation)- Alternative hypothesis is the compliment of the null hypothesis. It’s everything the null hypothesis is noto Lacks equality- Note that our conclusion is stated with regards to the null hypothesis, which can either be i.) rejected or ii.) not rejected. That is, we never accept the nullhypothesiso You can never prove the null is right because we are working with sample datao We are trying to refute the evidence against the null, not accepting it - We should not look at the data before conducting a test. That would color what kind of test we perform and bias the data- Critical Region: area containing all possible estimated values of the parameter that will result in rejection of the null hypothesis- Critical Value(s): the value(s) that divide the critical region(s) from the ‘do not reject’ region- Test Statistic: sample based value that will be compared to the critical region(s) to decide weather to reject or not reject the null hypothesis. The generic form of the test statistic is:- This is a Z-Score. We can use a standard normal table- Decision Rule: Statement that specifies values of the test statistic that will result in rejection or non-rejection of the null hypothesis-Two-Tailed Hypothesis: Evaluation of a conjecture for which sample results that are either significantly less than or greater than the conjectured value of the parameterwill result in rejection of the null hypothesis.o i.e., for a null hypothesis that only includes equality between conjectured vale and the tested parameter-One-Tailed Hypothesis: evaluation of a conjecture for which sample results are only sufficiently less than or greater than the conjectured value will result in rejection of the null hypothesiso i.e., for a null hypothesis that includes an inequality between the conjectured value and the test parameter-Upper-Tailed Test: Hypothesis test for which sample results are only sufficiently greater than the conjectured value of the parameter will result in rejection of the null hypothesis-Lower-Tailed Test: hypothesis test for which the sample results areonly sufficiently less than the conjectured value of the parameter will result in rejection of the null hypothesis-The only way we can know if the null hypothesis is true is if we take a census. We are taking a sample which are not perfectly representative of the population-Level of Significance: the probability of rejecting the null hypothesis at equality when it is actually true.-Type I Error: Rejection of a true null hypothesis. The probability of this occurrence (given the null hypothesis is true) is denoted as -Type II Error: non-rejection of a false null hypothesis. The probability of this occurrence (given the null hypothesis is false in some specific manner) is denoted as -Even if you do everything right, you can still have these errors. Sometimes samples don’t look like the population they were taken from- A note about Type I and II errors- the probability of a Type I error is actually a conditional probability- So we also have that- There is only one condition under which Type I can occur-the null hypothesis must be true- Note that for this null hypothesis we could write- So we also have:- The Probability of a Type II error is also a conditional probability that is frequently written as- So we also have- However, type II error in non-rejection of a false null hypothesis-there are an infinite number of ways in which this condition can exist- Thus, we usually discuss type II error in terms of a specific manner in which the null hypothesis is false- Finally note that the probability is referred to as the power of the hypothesis testSTEPS IN HYPOTHESIS TESTING- Based on the scientific method- The steps:1) State the null and the alternative hypothesis2) Select an appropriate test statistic3) State the desired level of significance a, find the critical value(s) and state the decision rule4) Calculate the test statistic from the sample data5) Use the decision rule to evaluate the test statistic and decide whether to reject or not reject the null hypothesis. Interpret your