The Hypothesis Testing Framework

Investigating questions and claims about population parameters.

�Objectives and Tools for Inferential Statistics

• Most recently, we’ve discussed how we can use confidence intervals to capture population parameters
• The result of constructing a confidence interval is a range of values, within which we are reasonably confident (90%, 95%, etc.) that the population parameter falls
• Purpose: These confidence intervals help us answer “What is the value of a population parameter?”
• We won’t always want to estimate the value of a population parameter; sometimes we’ll want to answer questions or investigate claims about it instead
• The hypothesis testing framework provides a strategy for doing this

�Examples of Questions/Claims to Investigate

• Is the average delivery weight of a one-ton order of wood stove pellets less than the 2000lbs promised?
• Is the proportion of granted requests for promotions at XYZ Home Heating below 25%?
• Is there evidence to suggest that the proportion of granted promotion requests differs by gender identity?
• Is there evidence to suggest that employee salaries at XYZ Home Heating increase during an individual’s first three years, after adjusting for inflation?
• Is there evidence to suggest a difference in customer satisfaction ratings between XYZ Home Heating’s Manchester and Plymouth offices.

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�Intuition for Testing Claims

Let’s begin with the accusation that a one-ton delivery of home heating pellets from XYZ Home Heating actually contains less than one ton (2000lbs) of pellets

• How might we investigate this claim?
• Collecting and weighing a random sample of pellet deliveries on its own is not sufficient, because we know that the average weight will vary from one sample to the next
• The burden of proof should be on the accusers, so we should begin from the assumption that a pellet delivery from XYZ Home Heating weighs at least the guaranteed 2000lbs
• We then collect and weigh our random sample of pellet deliveries and measure the likelihood that that sample came from a population whose mean weight is at least 2000lbs
• If our observed data is extremely unlikely (say samples like ours appear with less than 5% probability), then our sample is not compatible with a reality in which the average delivery weight is at least 2000lbs
• Otherwise, our observed data is consistent with the assumption, and the population may have an average weight of at least 2000lbs

�The Null and Alternative Hypotheses

Intermediate Decisions, Assumptions, and Calculations in Hypothesis Tests

Determining the Result of �the Test

�Examples

�Examples: Intramural Sport Participation

Scenario: Prior to the COVID-19 pandemic, a college reported that 35% of its students participate in intramural sports. They wonder if post-COVID participation in intramural sports is lower than it was pre-COVID. They conducted a test using statistical software, the results appear below.

success: yes �n = 100, p-hat = 0.31 �z = -0.8649 �p-value = 0.1936

Write out the hypotheses for the test and determine the result at the 5% level of significance, with justification.

�Example: Electric Scooter Range

Scenario: An electric scooter manufacturer claims their scooters can travel further than their leading competitor’s scooters after a full charge. The competitor’s scooters average a 25-mile range after charging to full capacity. The manufacturer conducts a test at the 10% level of significance, the results of which appear below.

n = 12, y-bar = 28.0865, s = 5.4596 �t = 1.9583, df = 11 �p-value = 0.038

Write out the hypotheses for the test and determine the result, with justification.

�Example: Streaming Platform Usage

�Example: Streaming Service Subscriptions

�Errors in Statistical Inference

In using statistical inference, there is no guarantee that the conclusions we arrive at are correct

• We are dealing with imperfect information, resulting from random sampling and noisy data

The Null Hypothesis Significance Testing (NHST) methods we are utilizing result in one of four scenarios:

�Example: Manufacturing

Scenario: A company makes custom bolts that are used in a specialized manufacturing process. The bolts produced in the process are not perfectly identical – there is some slight variation in the length of a completed bolt. Bolts that are more than 1.5mm too long or too short cannot be used in the manufacturing process and must be discarded. Engineers have determined that as long as the average length of a produced bolt is 3.25”, then nearly all bolts are compliant. After receiving complaints, the company wonders whether the average length of a manufactured bolt is no longer 3.25”.

Write the hypotheses involved in a test of average bolt length. Discuss what a Type I error is in this context, as well as what a Type II error is. What are the consequences of each, and which one is more severe?

�Summary

�Next Time…

• What we’ll be doing…
• Hypothesis Tests for a Single Population Mean
• How to prepare…
• Read sections 8.4 and 8.5 in our textbook
• Homework: Start HW 7 (Hypothesis Tests for Parameters of a Single Population) on MyOpenMath