Hypothesis Tests for a Single Population Proportion
Investigating whether a population proportion is greater than, less than, or has changed from a previously accepted, assumed, or required value
�A Reminder on Proportions
�Good News…
�Examples
�Example: Smartphone Ownership
Scenario: A 2015 study on technology adoption in a city claimed that 85% of adults own a smartphone. In a recent random survey of 507 adults, 446 report owning a smartphone. Conduct a test to determine whether the proportion of smartphone ownership now differs from 85%.
�Example: Candidate Support
Scenario: A political candidate claims that 55% of voters in their district support them. A potential challenger has formed an exploratory committee and will only run if the candidate’s true level of support is below 55%. The chair of the committee takes a poll of 417 randomly selected voters from that district, showing that 223 of them support the candidate. Conduct a test at the 10% level of significance to determine whether the candidate should run for election.
�Example: Data Analyst Salaries
Scenario: A national labor report claims that the average salary for data analysts is $58,000 per year. An audit of 45 randomly selected data analysts at Datascope.io finds that their average salary is $59,500 with a standard deviation of $4,200. Conduct a test to determine whether the audit suggests that the average data analyst’s salary at Datascope.io is different from the national average.
�Example: Emissions Tests
Scenario: Historically, 95.6% of vehicles in Springfield pass the annual emission test. The state EPA is investigating whether Quick-Fix Auto (the auto repair shop famous for their tag line: “If you got money, we got you…no questions asked!”) is passing vehicles at a higher than expected rate. A random sample of 335 vehicles tested at Quick-Fix shows that 329 passed. Does this sample provide significant evidence that the proportion of vehicles passing their emissions inspections at Quick-Fix Auto is higher than the 95.6% expected?
�Example: Hospital Stay Lengths
Scenario: A hospital claims that the average length of stay for a patient undergoing minor surgery is 3.2 days. A sample of 60 recent patients shows an average stay of 3.5 days with a standard deviation of 0.9 days. Is there evidence suggesting that the average length of a hospital stay has increased?
�Example: Online Shopping and Impulse Purchases
Scenario: An online store claims that 59.3% of its shoppers make impulse purchases when shown suggested products at checkout (a strategy known as up-selling). After surveying 417 recent shoppers, 236 report making an impulse purchase. Conduct a test to determine whether the sample provides evidence of a change in the success rate of their up-selling strategy.
Example: Prescription Medication Dosing �Regimen Adherence
Scenario: A research article from 2020 claims that 83.9% of patients follow their prescribed medication regimen. A doctor suspects that proportion of their patients following dosing guidelines is less than that benchmark. After interviewing 117 of their patients, the doctor records that 94 are following their medication guidelines. Conduct a test to determine whether the sample data provides evidence to confirm the doctor’s suspicion.
�Examples: Defective Car Doors
Scenario: A factory claims that only 2% of car doors manufactured along a particular assembly line are defective. After a random sample of 800 doors is inspected, it is found that 25 of them are defective. Conduct a hypothesis test at the 1% level of significance to assess whether the defect rate is greater than the claimed 2%.
�Example: Cable Provider Satisfaction
Scenario: A major cable provider claims that 90% of customers are satisfied with their service. After randomly sampling 150 customers, 130 report that they are satisfied. Conduct a test to determine whether customer satisfaction is actually below the 90% claimed.
�Example: Software Adoption
Scenario: A company introduced a new software application three months ago and claims that 70% of employees now use it daily. After surveying 250 employees, it is found that 175 use the software daily. Conduct a test at the 2% level of significance to determine whether the actual usage rate is significantly less than 70%.
�Comments
�Next Time…