Hypothesis Tests for a Single Population Mean
Investigating whether a population mean is greater than, less than, or has changed from a previously accepted, assumed, or required value
�Warm-Up Scenario: Lightbulb Lifespan
Scenario: DimWatt Lighting, a discount lightbulb manufacturer advertises that their incandescent lightbulbs have an average lifetime of 1000 hours. After receiving multiple complaints, a consumer protection group investigates whether DimWatt Lighting is engaging in false advertising. The group purchases a random sample of 63 lightbulbs and observe an average lifespan of about 975.65 hours. The result of their test appears below.
n = 63, y-bar = 975.6487, s = 144.3857 �t = -1.3387, df = 62 �p_value = 0.0928
Using the 5% level of significance, determine whether this sample provides evidence to suggest that DimWatt Lighting’s bulbs have a lifespan below 1000 hours. Write out the hypotheses for the test and determine the result of the test.
Additional Uncertainty with Inference On Means:�A Reminder
�A General Strategy for Hypothesis Testing
In our previous discussion, we built some intuition for hypothesis testing
You can find a full strategy document here; remember that you’ll need the Standard Error Decision Tree to help you as well
Before we jump into completing examples, the steps involved in conducting a hypothesis test include:
�Example: Fitness Tracker Accuracy
Scenario: A fitness company claims that its new fitness tracker accurately measures the number of steps walked per day with an average error of no more than 200 steps. To test this, a researcher collects data from a sample of 12 users, finding that the average error in their recorded steps is 258, with a standard deviation of 57 steps. The company wants to know if the tracker is significantly less accurate than advertised. Conduct a hypothesis test to evaluate whether the tracker’s error in step count exceeds the stated threshold. Use the 5% level of significance.
�Example: Fitness Tracker Accuracy
�Example: Fitness Tracker Accuracy �
�Example: Fitness Tracker Accuracy
�Example: Fitness Tracker Accuracy
�Example: Fitness Tracker Accuracy
�Example: Fitness Tracker Accuracy
Decision in Context:
The sample data provided significant evidence that the company’s claim is wrong. The average error in step-count exceeds 200.
�Ask Me Two Questions
Ask me at least two questions before we move forward
�Example: Medication Effectiveness
�A Comment on the Previous Example
�Example: Flight Delays
Scenario: An airline asserts that its flights are on time 90% of the time, with an average delay of no more than 15 minutes. A researcher randomly selects 100 flights from the past month and finds that the average delay is 17 minutes with a standard deviation of 5 minutes. The researcher performs a hypothesis test to see if the average delay exceeds the claimed 15-minute threshold. Use a 10% level of significance.
�Example: Smartphone Charging Times
Scenario: A smartphone brand claims that its battery takes, on average, 90 minutes to fully charge. A sample of 55 users report an average charging time of 97.5 minutes with a standard deviation of 11.2 minutes. Conduct a hypothesis test to evaluate if the average charging time is significantly longer than the claimed 90 minutes.
�Example: Electric Car Battery Life
Scenario: An electric car manufacturer advertises that the average lifespan of its car batteries is 8 years. A study is conducted with a sample of 40 electric cars, and the average battery lifespan is found to be 7.6 years with a standard deviation of 0.9 years. Conduct a hypothesis test to assess whether the average battery life is significantly shorter than the advertised 8 years.
�Example: Coffee Shop Wait Times
Scenario: A popular coffee shop prides itself on quick service, aiming to serve customers in an average time of 3.5 minutes. After receiving customer complaints about long wait times, the manager decides to test if the average wait time has exceeded this target. A random sample of 45 customers is observed, and the mean wait time is found to be 3.8 minutes with a standard deviation of 0.6 minutes. Conduct a hypothesis test to determine if the average wait time is significantly greater than 3.5 minutes.
�Comments
�Next Time…