Inference for Comparing Two Population Means

Investigating whether a mean differs across two populations and estimating that difference

�A Reminder

What’s New? �Thankfully, not much!�

�Examples

• As usual, the following slides contain example problems for us to complete
• Each example involves a request to perform inference comparing means across two populations
• You’ll need to decide…
• What type of inference is appropriate (confidence interval or hypothesis test)
• One more thing…paired data…
• Two populations are paired if there is a natural and direct comparison between each individual observation in one sample with an observation in the other sample
• One common scenario where paired data arises is when the same participant is measured at two different points in time (for example, blood pressure before and after taking a medication)
• In these cases, we compute the differences in the measurements and run inference on the differences

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�Example: Game Score by Console

Scenario: A gaming company compares the average scores of players using two different gaming consoles, PS5 and XBOX Series X. A random sample of 50 players on the PS5 scores an average of 87.4 (SD = 5.6), while 48 players on XBOX Series X score an average of 89.7 (SD = 4.9). Is there evidence to suggest inconsistent scoring between the two systems?

�Example: Reaction Times Before and After Coffee

Scenario: A researcher tests whether drinking coffee improves reaction times. A sample of 20 college students completes a reaction time task both before and 30 minutes after consuming coffee. The mean difference in reaction times is 0.38 seconds faster after coffee with a standard deviation of 0.14 seconds. Conduct a hypothesis test at the 10% level of significance to determine whether the improvement is statistically significant.

�Example: Effectiveness of Study Groups

Scenario: A professor for a large lecture course identifies that some students have decided to form study groups while others have decided to study alone. After a recent exam, the professor collected a random sample of exams including 23 students who studied in groups and 25 students who studied alone. The mean score for students in study groups was 78.2 (SD = 7.3), while the students who studied alone averaged 74.6 (SD = 6.8). Construct a 95% confidence interval for the effect that studying in a group has on exam performance.

�Example: Calories Burned by Fitness Class

Scenario: A fitness center compares calories burned during a cycling class versus a kickboxing class. A random sample of 31 participants in the cycling class burns an average of 450 calories with a standard deviation of 48 calories, while 28 participants in the kickboxing class burn an average of 480 calories with a standard deviation of 53 calories. Conduct a test to determine whether there is evidence to suggest that kickboxing burns more calories than cycling.

�Example: Sleep Duration in Singles versus Doubles

Scenario: A sleep study investigates whether students in single-occupancy dorms sleep longer than those in shared dorms. In a sample of 29 students from single rooms, the average sleep duration is 7.8 hours with a standard deviation of 0.9 hours, while in 36 students from shared rooms, it is 7.4 hours with a standard deviation of 1.2 hours. Construct a 98% confidence interval for the difference in average sleep durations.

�Example: Test Anxiety

Scenario: A psychologist studies whether high school students experience greater test anxiety than college students. A sample of 32 high school students scores an average of 4.7 on a 10-point anxiety scale with a standard deviation of 1.8 points, while 37 college students score an average of 4.2 with a standard deviation of 1.6. Do the data provide evidence to suggest greater anxiety in high school test-takers?

�Example: Entertainment Spending

Scenario: A survey investigates whether urban and rural teens have different entertainment spending habits. In a sample of 28 urban teens, the average monthly spending is $83.48 with a standard deviation of $21.73, while 31 rural teens spend an average of $74.26 with a standard deviation of $18.97. Does the observed data provide evidence of a difference in average monthly entertainment spending?

�Example: Weight Loss and Diet

Scenario: A nutritionist compares weight loss between a keto diet and intermittent fasting. For the keto diet, a sample of 45 participants loses an average of 12.3lbs in 12 weeks with a standard deviation of 4.8lbs. For intermittent fasting, 38 participants lose an average of 10.7lbs with a standard deviation of 5.1lbs. Construct a 95% confidence interval for the difference in average weight loss between the two diets.

Example: Screen Time on Weekdays versus Weekends

Scenario: A researcher is interested in whether high school students spend more time on their phones during weekends compared to weekdays. A random sample of 42 students reports an average screen time of 5.2 hours on weekdays (SD = 1.1) and 6.7 hours on weekends (SD = 1.4). The average difference in screen time (weekend – weekday) for each student was 1.3 hours with a standard deviation of 0.74 hours. Test whether there is a significant difference in screen time for high school students on weekdays versus weekends.

Inference: Where We’ve Been and Where �We Are Headed

�Next Time…

• What we’ll be doing…
• Inference for Comparisons of Two Population Proportions
• How to prepare…
• Read section 9.5 in our textbook
• Homework: Begin HW 8 (Hypothesis Tests for Comparing Parameters Across Two Populations) on MyOpenMath