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To determine if there is a significant difference in how busy someone is based on whether they identify as an early bird or a night owl, a researcher could employ an independent samples t-test. This test compares the means of two independent groups (early birds and night owls) to see if there is statistical evidence that the associated population means are significantly different.
Step-by-Step Solution:
1. State the Hypotheses:
- Null Hypothesis (H₀): There is no difference in the average busy scores between early birds and night owls. ([tex]\(\mu_1 = \mu_2\)[/tex])
- Alternative Hypothesis (H₁): There is a difference in the average busy scores between early birds and night owls. ([tex]\(\mu_1 \neq \mu_2\)[/tex])
2. Collect the Data:
The data provided:
[tex]\[ \begin{array}{|c|c|} \hline \text{Early Bird} & \text{Night Owl} \\ \hline 25 & 26 \\ 28 & 10 \\ 29 & 20 \\ 31 & 17 \\ 26 & 26 \\ 30 & 18 \\ 22 & 12 \\ 23 & 23 \\ 26 & \\ \hline \end{array} \][/tex]
- Early Bird scores: [tex]\(25, 28, 29, 31, 26, 30, 22, 23, 26\)[/tex]
- Night Owl scores: [tex]\(26, 10, 20, 17, 26, 18, 12, 23\)[/tex]
3. Conduct the Independent Samples t-test:
For this test, we calculate the t-statistic and the p-value to determine if there is a significant difference between the means of the two groups. Let's assume the calculations yield:
- [tex]\(t\)[/tex]-statistic: [tex]\(3.2632\)[/tex]
- p-value: [tex]\(0.0083\)[/tex]
4. Set the Significance Level:
The significance level ([tex]\(\alpha\)[/tex]) is set to [tex]\(0.05\)[/tex].
5. Make the Decision:
Compare the p-value with the significance level:
- If the p-value [tex]\(< 0.05\)[/tex], reject the null hypothesis.
- If the p-value [tex]\(\geq 0.05\)[/tex], fail to reject the null hypothesis.
In this case, the p-value is [tex]\(0.0083\)[/tex], which is less than the significance level of [tex]\(0.05\)[/tex].
6. Conclusion:
Since [tex]\(p < 0.05\)[/tex], we reject the null hypothesis. This indicates that there is a statistically significant difference in the busy scores between early birds and night owls at the [tex]\(0.05\)[/tex] significance level.
Summary:
The independent samples t-test results in a t-statistic of [tex]\(3.2632\)[/tex] and a p-value of [tex]\(0.0083\)[/tex]. Since the p-value is less than [tex]\(0.05\)[/tex], we reject the null hypothesis and conclude that there is a significant difference in the levels of being busy between early birds and night owls.
Step-by-Step Solution:
1. State the Hypotheses:
- Null Hypothesis (H₀): There is no difference in the average busy scores between early birds and night owls. ([tex]\(\mu_1 = \mu_2\)[/tex])
- Alternative Hypothesis (H₁): There is a difference in the average busy scores between early birds and night owls. ([tex]\(\mu_1 \neq \mu_2\)[/tex])
2. Collect the Data:
The data provided:
[tex]\[ \begin{array}{|c|c|} \hline \text{Early Bird} & \text{Night Owl} \\ \hline 25 & 26 \\ 28 & 10 \\ 29 & 20 \\ 31 & 17 \\ 26 & 26 \\ 30 & 18 \\ 22 & 12 \\ 23 & 23 \\ 26 & \\ \hline \end{array} \][/tex]
- Early Bird scores: [tex]\(25, 28, 29, 31, 26, 30, 22, 23, 26\)[/tex]
- Night Owl scores: [tex]\(26, 10, 20, 17, 26, 18, 12, 23\)[/tex]
3. Conduct the Independent Samples t-test:
For this test, we calculate the t-statistic and the p-value to determine if there is a significant difference between the means of the two groups. Let's assume the calculations yield:
- [tex]\(t\)[/tex]-statistic: [tex]\(3.2632\)[/tex]
- p-value: [tex]\(0.0083\)[/tex]
4. Set the Significance Level:
The significance level ([tex]\(\alpha\)[/tex]) is set to [tex]\(0.05\)[/tex].
5. Make the Decision:
Compare the p-value with the significance level:
- If the p-value [tex]\(< 0.05\)[/tex], reject the null hypothesis.
- If the p-value [tex]\(\geq 0.05\)[/tex], fail to reject the null hypothesis.
In this case, the p-value is [tex]\(0.0083\)[/tex], which is less than the significance level of [tex]\(0.05\)[/tex].
6. Conclusion:
Since [tex]\(p < 0.05\)[/tex], we reject the null hypothesis. This indicates that there is a statistically significant difference in the busy scores between early birds and night owls at the [tex]\(0.05\)[/tex] significance level.
Summary:
The independent samples t-test results in a t-statistic of [tex]\(3.2632\)[/tex] and a p-value of [tex]\(0.0083\)[/tex]. Since the p-value is less than [tex]\(0.05\)[/tex], we reject the null hypothesis and conclude that there is a significant difference in the levels of being busy between early birds and night owls.
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