The problem describes a study where a researcher wants to determine if there is a significant difference in the "busyness" scores between people who identify as "early birds" and those who identify as "night owls". The data provided represents the busyness scores for each group. The researcher is testing for a difference at the $\alpha = 0.05$ level of significance. To analyze this data, we would typically conduct an independent samples t-test (assuming the data meets the assumptions for this test, such as normality and equal variances). However, we are only asked to set up the problem. We are not asked to conduct the t-test.

1. Problem Description

The problem describes a study where a researcher wants to determine if there is a significant difference in the "busyness" scores between people who identify as "early birds" and those who identify as "night owls". The data provided represents the busyness scores for each group. The researcher is testing for a difference at the $\alpha = 0.05$ level of significance. To analyze this data, we would typically conduct an independent samples t-test (assuming the data meets the assumptions for this test, such as normality and equal variances). However, we are only asked to set up the problem. We are not asked to conduct the t-test.

2. Solution Steps

First, define the null and alternative hypotheses:

$H_0$: There is no difference in the mean busyness scores between early birds and night owls. ($\mu_{EarlyBird} = \mu_{NightOwl}$)

$H_1$: There is a difference in the mean busyness scores between early birds and night owls. ($\mu_{EarlyBird} \neq \mu_{NightOwl}$)

Second, list the data:

Early Bird: 25, 28, 29, 31, 26, 30, 22, 23, 26

Night Owl: 26, 10, 20, 17, 26, 18, 12, 23

Third, since the problem does not ask us to conduct the full hypothesis test, we stop at setting up the problem. We could compute sample means, sample standard deviations, and then conduct a t-test for independent samples.

3. Final Answer

Null Hypothesis ($H_0$): $\mu_{EarlyBird} = \mu_{NightOwl}$

Alternative Hypothesis ($H_1$): $\mu_{EarlyBird} \neq \mu_{NightOwl}$

Early Bird Data: 25, 28, 29, 31, 26, 30, 22, 23, 26

Night Owl Data: 26, 10, 20, 17, 26, 18, 12, 23

Significance Level ($\alpha$): 0.05