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UTPA STEM/CBI Courses/Probability and Statistics/Statistical Inference on Mean

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Course Title: Elementary Statistics and Probability

Lecture Topic: Statistical Inference on Mean through Tests of Hypotheses

Instructor: Dr. Santanu Chakraborty

Institution: University of Texas - Pan American

Backwards Design

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Course Objectives

  • Primary Objectives- By the next class period students will be able to:
    • Have a basic idea of estimation and tests of hypotheses for a population mean
    • Be able to calculate the test statistic using the given information
    • Be able to make a decision either in favor of the claim or against the claim in a given problem for a given level of significance using p-value approach or critical value approach


  • Sub Objectives- The objectives will require that students be able to:
    • Connect it to the previous chapter and reason out the distribution of the sample mean using Central Limit Theorem
    • Recapitulate the previous knowledge of calculating sample mean and sample standard deviation if needed for a given data set


  • Difficulties- Students may have difficulty:
    • In identifying the correct null hypothesis and the alternative hypothesis for a word problem
    • Interpreting the sample mean, sample size, population mean, etc., in a word problem and substitute them correctly in the formula for the test statistic
    • Interpret decisions made in terms of the language of the problem


  • Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
    • If a new drug is to be tested to see if it has any effect in controlling the blood pressure, data can be collected on patients who have already used it. This can be compared with the average number for the patient population who do not use it through a test of hypothesis
    • If CBI is to be tested to determine whether it has improved the teaching evaluation of an instructor for a particular course, his/her teaching evaluation for a CBI-based teaching course can be compared with the average teaching evaluations for non-CBI based teaching courses


Model of Knowledge

  • Concept Map
    • Concepts of Null and Alternative Hypotheses
    • Concepts of Errors that one makes while deciding to be in favor of null or alternative hypotheses
    • The main part of the analysis which involves calculating the test statistic and using p-value or critical value approach to make a decision


  • Content Priorities
    • Enduring Understanding
      • Understanding Null and Alternative Hypotheses
      • Interpret the decision in the language of the problem


  • Important to Do and Know
    • Know the formula of the test statistic
    • Look at the appropriate tables for the p-value and critical approaches
    • What it means to say "reject the null hypothesis" or "fail to reject the null hypothesis" at a given level of significance using p-value and critical value approaches


  • Worth Being Familiar with
    • Difference between sample mean and population mean
    • Notations for sample mean, sample variance, population mean and population variance
    • How to calculate sample mean and sample standard deviation
    • Central Limit theorem


Assessment of Learning

  • Formative Assessment
    • In Class (groups)
      • Identifying null and alternative hypotheses in some given word problems in the book
      • If the decision is to reject the null hypothesis in a given word problem in the book, what should the interpretation in the language of the problem be?


  • Homework (individual)
    • Calculate the test statistic given in some problems posted on Blackboard.
    • Calculative the p-values and critical values in some problems posted on Blackboard.


  • Summative Assessment
    • Take an individual exam over tests of hypotheses for mean in class.
    • Give a presentation on a problem in your major where you apply tests of hypotheses for mean.


Legacy Cycle

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OBJECTIVE

By the next class period, students will be able to:

  • Identify the null and alternative hypotheses in a given problem
  • Calculate the test statistic and use p-value or critical value approach to make a decision
  • Interpret the decision in the language of the problem


The objectives will require that students be able to:

  • Determine the pair of hypotheses and identify the parameters involved
  • Be very familiar with the formula of the test statistic and be able to use the appropriate tables for p-value approach and critical value approach
  • Come up with challenging questions.
    • An example is: "What is to be done in a practical situation when you reject the null hypothesis under some level of significance and fail to reject for another level?"



THE CHALLENGE You have the data on the systolic blood pressure of 100 patients who are using a particular drug to keep their blood pressure under control. The average blood pressure calculated for these patients is 125 and the variance is 144. From your experience, you know that the average blood pressure for similar patients who is not using the drug is 130. Can you claim that the new drug has controlled the blood pressure for these patients at a 5% level of significance?



GENERATE IDEAS

Students may ask the following:

  • "How do I formulate the claim in a statistical language?"
  • "How is our previous knowledge in mathematics connected with the present problem?"
  • "I understand what a sample mean and population mean are. I see several averages in the problem. What is sample and population mean in this problem?"
  • "I know some inference already. I know how to calculate the confidence interval for a population mean. Is it related to this problem?"
  • "What is meant by the level of significance?"



MULTIPLE PERSPECTIVES

  • The null hypothesis and the alternative hypothesis are given in order to formulate the claim. One of them supports the claim and the other is a negation of that.
  • As mentioned earlier, the sample mean is used to conclude in the population mean. So, all the knowledge with respect to data summarizing, data analysis, and probability theory come into the picture. The sample mean and the sample variance is calculated if it is not provided. The Central Limit Theorem is used to discuss the probability distribution of the sample mean.
  • The average that is calculated based on the sample is the sample mean. So, 125 is the sample mean. The average that is not based on any sample is the population mean. It is the patient population who are not using the drug.
  • It is related to the confidence interval. In fact, the confidence interval can be used to test the claim. But here different techniques will be used like p-value approach and classical approach.
  • At the end of the analysis, a decision can be made. If an error is made by rejecting the null hypothesis when it is correct, the result will be an error. The level of significance is the probability of making such an error.



RESEARCH & REVISE

  • The null and alternative hypothesis is described. The students will be given the languages to write them down. Groups will be created in the class and they will be asked to study identifying null and alternative hypothesis. They will use select problems from the book to do this.
  • Then they will be shown through a step-by-step method given in the book/webpage/projector how to do the analysis using both the test statistic formula, and p-value and critical value approaches.
  • The students will return to their groups to interpret some of the results in the book problems after a decision is taken in favor of against a claim made in a problem.
  • Homework will be assigned to do some tests of hypotheses problems. This will involve going through all the steps.
  • Students will be given an opportunity to enhance their learning. Some references will be provided to the students so that they have different perspectives.


TEST YOUR METTLE

  • A discussion will occur over the results of the group study of identifying null and alternative hypotheses and interpretation of results.
  • Students to turn in their homework. A discussion over this homework will occur in the next class.


GO PUBLIC

  • After the students receive feedback on their work, they take an in-class, individual test.
  • Once they have a grasp on the subject, they will be given group presentations. Expected from these group presentations will be how to apply their new-found knowledge in their majors like nursing, biology, communication, psychology, criminal justice etc..
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