Thank you for visiting Listed in the accompanying table are waiting times in seconds of observed cars at a Delaware inspection station The data from two waiting lines are. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
To solve this problem, we'll go through the steps provided and understand the results derived from the problem statement. We are comparing waiting times from two different setups at an inspection station: a single waiting line and two waiting lines. Here's how to interpret the problem:
### Part (a): Hypothesis Testing
1. Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The means of the two populations ([tex]\(\mu_1\)[/tex] and [tex]\(\mu_2\)[/tex]) are equal ([tex]\(\mu_1 = \mu_2\)[/tex]).
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The means of the two populations are not equal ([tex]\(\mu_1 \neq \mu_2\)[/tex]).
2. Test Statistic:
- The test statistic calculated is approximately [tex]\(t = 0.16\)[/tex]. This statistic measures the standardized difference between the means of the two samples.
3. P-value:
- The P-value found is approximately [tex]\(0.874\)[/tex]. The P-value helps us determine the strength of the evidence against the null hypothesis.
4. Conclusion:
- We typically set a significance level (commonly 0.05 for 95% confidence) to decide whether to reject the null hypothesis.
- Since the P-value is [tex]\(0.874\)[/tex], which is greater than the significance level of 0.05, we fail to reject the null hypothesis.
- Conclusion: There is not enough evidence to conclude that the waiting times for a single line are different from two waiting lines.
### Part (b): Confidence Interval
1. Confidence Interval for the Difference in Means ([tex]\(\mu_1 - \mu_2\)[/tex]):
- A 95% confidence interval for the difference in means is calculated to further interpret the results.
- The confidence interval is: [tex]\(-140.5 < \mu_1 - \mu_2 < 165.4\)[/tex].
2. Interpretation of the Confidence Interval:
- This interval includes zero, suggesting that there is no significant difference in the average waiting times between the two setups.
By going through these steps, we've analyzed the waiting time data to understand the differences (or lack thereof) between having a single line versus two lines at an inspection station.
### Part (a): Hypothesis Testing
1. Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The means of the two populations ([tex]\(\mu_1\)[/tex] and [tex]\(\mu_2\)[/tex]) are equal ([tex]\(\mu_1 = \mu_2\)[/tex]).
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The means of the two populations are not equal ([tex]\(\mu_1 \neq \mu_2\)[/tex]).
2. Test Statistic:
- The test statistic calculated is approximately [tex]\(t = 0.16\)[/tex]. This statistic measures the standardized difference between the means of the two samples.
3. P-value:
- The P-value found is approximately [tex]\(0.874\)[/tex]. The P-value helps us determine the strength of the evidence against the null hypothesis.
4. Conclusion:
- We typically set a significance level (commonly 0.05 for 95% confidence) to decide whether to reject the null hypothesis.
- Since the P-value is [tex]\(0.874\)[/tex], which is greater than the significance level of 0.05, we fail to reject the null hypothesis.
- Conclusion: There is not enough evidence to conclude that the waiting times for a single line are different from two waiting lines.
### Part (b): Confidence Interval
1. Confidence Interval for the Difference in Means ([tex]\(\mu_1 - \mu_2\)[/tex]):
- A 95% confidence interval for the difference in means is calculated to further interpret the results.
- The confidence interval is: [tex]\(-140.5 < \mu_1 - \mu_2 < 165.4\)[/tex].
2. Interpretation of the Confidence Interval:
- This interval includes zero, suggesting that there is no significant difference in the average waiting times between the two setups.
By going through these steps, we've analyzed the waiting time data to understand the differences (or lack thereof) between having a single line versus two lines at an inspection station.
Thank you for reading the article Listed in the accompanying table are waiting times in seconds of observed cars at a Delaware inspection station The data from two waiting lines are. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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