Week 10 - Putting It All Together with Inferential Statistics

Why

Recall that one of our original goals was to use a sample mean to estimate a population mean. We will now formulate the error for this estimation and we can then make scientific arguments regarding the range of values that a population mean can take on. This will show how the sample size, spread, and confidence of an estimate all play an important role in estimation. Finally you will know enough to validate the statistical claims made by others about population means. The process will also be applied to proportions but the theory behind these estimates will not be developed.

Learning Objectives

• Know how to determine and interpret a confidence interval for the mean
• Know how to determine and interpret a confidence interval for the proportion
• Understand the roles of sample size, confidence level, and spread on the margin of error.

Performance Criteria

• The learner will calculate confidence intervals using technology accurate to 2 decimal places.
• The learner will interpret confidence intervals using a complete sentence that includes the units of the random variable and the confidence level.

Why

The culmination of all our hard work so far this semester will by the study of hypothesis testing. In this experience we will examine each part of the process both conceptually and as a set of calculations. Thorough understanding of these mechanics will ensure you do not make the common mistakes that plague the uninformed. This important procedure is the industry standard technique for scientifically supporting and refuting claims about population means and proportions. You will find that something similar to these tests is used to determine if drugs are safe, new methods are better, or if people's opinions have changed. 

Learning Objectives:

• Know the general process of hypothesis testing using p-values
• Learn how to write hypotheses
• Learn how the decision is used to make a formal conclusion.

Performance Criteria

• The learner will translate written scenarios into corresponding null and alternative hypotheses using correct notation for mean and proportion tests.
• The learner will translate written scenarios and small data sets into corresponding hypothesis tests and use the complete procedure to test claims.
• The learner will write a conclusion in complete sentences that relates the decision to the original hypotheses and includes the level of significance.

Chapter 8, section 1-3

Chapter 9, section 1-5, especially section 5

Chapter 8 This is a video by the authors covering chapter 8 

Confidence Intervals for the Mean (TI-84)

Confidence Intervals for the Mean (Excel)

Confidence Intervals for the Proportion (TI-84)

Confidence Intervals for the Proportion (Excel)

Chapter 9 This is a video by the authors covering chapter 9.

 Calculating P-Values (Excel/Sheets)

 Hypothesis Testing - Z-test (Excel/Sheets)

 Hypothesis Testing - t-test (Excel/Sheets)

 Hypothesis Testing - proportion-test (Excel/Sheets)

 Calculating P-Values (TI-84)

 Hypothesis Testing - Z-test (TI-84)

 Hypothesis Testing - t-test (TI-84)

 Hypothesis Testing - proportion-test (TI-84)

Use this model to write the conclusion in the Application.

The class will work together to create a wiki for inferential statistics. I will provide an outline and each of you will add at least one point (one or two complete sentences). Use the following guidelines

• Type your name in parentheses next to your contributions
• Try and keep it general - no examples.
• Check to make sure you are not repeating someone else.
• If you correct someone else, use strike-through (example) instead of deleting what they wrote. 
• Put things in your own words, don't just copy from the book.
• Make sure you put your response in the right place, use my outline as the guide.
• Use correct symbols or formulas where necessary. The Math Symbols button is available for you to use.

Your contribution to the Wiki is due by June 7, 2019

Suppose you are a researcher in a hospital. You are experimenting with a new tranquilizer. You collect data from a random sample of 8 patients. The period of effectiveness of the tranquilizer for each patient (in hours) is as follows:

1. What is a point estimate for the population mean length of time. (Round answer to 3 decimal places) 
   
   
2. Construct a 90% confidence interval for the population mean length of time. Enter your answer as an open-interval (i.e., parentheses) Round upper and lower bounds to three decimal places. 
   
   
3. Interpret the confidence interval in a complete sentence. Make sure you include units 
   
   
4. What does it mean to be "90% confident" in this problem? Use the definition of confidence level. 
   
   • There is a 90% chance that the confidence interval contains the population mean.
   • 90% of all simple random samples of size 8 from this population will result in confidence intervals that contain the population mean.
   • The confidence interval contains 90% of all samples.
5. Suppose that the company releases a statement that the mean time for all patients is 2 hours.
   1. Is this possible?
      •  
   2. Is it likely?
      •  
6. Use the results above and make an argument in favor or against the company's statement. Structure your essay as follows:
   1. Describe the population and parameter for this situation.
   2. Describe the sample and statistic for this situation.
   3. Give a brief explanation of what a confidence interval is.
   4. Explain what type of confidence interval you can make in this situation and why.
   5. Interpret the confidence interval for this situation.
   6. Restate the company's claim and whether you agree with it or not.
   7. Use the confidence interval to estimate the likelihood of the company's claim being true.
   8. Suggest what the company should do.