10. Hypothesis Testing for Two Samples
Two Means - Unknown, Unequal Variance
1:47 minutesProblem 3
Textbook Question
Randomization with Commute Times Given the two samples of commute times (minutes) shown here, which of the following are randomizations of them?
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a. Boston: 10 10 60. New York: 5 20 25 30 45.
b. Boston: 10 10 60 20 25. New York: 5 30 45.
c. Boston: 5 10 25 25 60. New York: 5 30 30 60.
d. Boston: 10 10 60. New York: 5 20 25 30 45.
e. Boston: 10 10 10 10 10. New York: 60 60 60.
Verified step by step guidance1
Step 1: Understand the concept of randomization. Randomization involves rearranging or redistributing the data points between groups while maintaining the total number of data points in each group. This process ensures that the groups are formed randomly without any inherent bias.
Step 2: Analyze the original samples provided. The original commute times are: Boston: 10, 10, 60 and New York: 5, 20, 25, 30, 45. These are the baseline distributions of commute times for the two cities.
Step 3: Evaluate each option to determine if it represents a valid randomization. For a valid randomization, the total number of data points in each group must remain consistent with the original sample sizes (Boston: 3 data points, New York: 5 data points), and the data points must be redistributed randomly between the groups.
Step 4: Check for duplicates or patterns that might indicate non-randomization. For example, in option (e), Boston has all identical values (10, 10, 10, 10, 10), and New York has all identical values (60, 60, 60). This is not a randomization but rather a systematic grouping.
Step 5: Compare each option to the original samples and verify whether the redistribution of data points adheres to the principles of randomization. Ensure that the total number of data points and the original values are preserved in the new groupings.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Randomization
Randomization is a statistical technique used to assign individuals or items to different groups in a way that eliminates bias. In the context of comparing two samples, randomization ensures that the samples are representative of the population, allowing for valid conclusions about differences or similarities. It is crucial for ensuring that the results of an analysis are not influenced by external factors.
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Sample Size and Composition
Sample size refers to the number of observations or data points collected in a study. The composition of a sample, including its distribution and range of values, affects the reliability of statistical analyses. In the given question, understanding the sample sizes and their respective values is essential to determine if the proposed randomizations maintain the integrity of the original samples.
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Statistical Validity
Statistical validity refers to the extent to which the conclusions drawn from a statistical analysis are accurate and applicable to the real world. This includes ensuring that the samples used are appropriate for the analysis being conducted. In the context of the question, assessing whether the proposed randomizations preserve the original sample characteristics is key to determining their validity.
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