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Chapter 10: Problem 142

A chemist is testing a proposed analytical method and has no established standards to compare it with, so she decides to use the currently accepted method for comparison. She takes a specimen of unknown concentrate and determines its concentration 12 times using the proposed method. She then takes another specimen of same unknown concentrate and determines its concentration 12 times using the current method. Do these two samples represent dependent or independent samples? Explain.

Short Answer

Expert verified
The two samples represent independent samples. This is because the measurements taken with the proposed method and the current method are not affected by each other and there is no natural pairing between the two sets of measurements.

Step by step solution

01

Understand Independent and Dependent Samples

In statistics, independent samples are those samples that do not affect each other. That is, the selection of one sample does not in any way influence, either by changing, controlling, or categorizing, the selection of another sample. On the other hand, dependent samples are those samples that have a natural pairing. They are generally used when comparing and finding differences within groups or between time periods.
02

Analyze the Sample Collection Process

In this problem, the chemist tests a specimen of unknown concentrate 12 times using the proposed method. This process is then repeated with a similar specimen using the current method. The samples come from a similar source, but the measurements taken are based on two entirely different methods.
03

Determine the Sample Type Based on the Procedure

Given that the measurements taken by the proposed method and the current method do not affect each other and there is no natural pairing between the two sets of measurements, it can be concluded that the two samples represent independent samples.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dependent Samples
Dependent samples in statistics are samples that have a natural, intrinsic connection or pairing. This often occurs when measurements are taken on the same subjects under different conditions or across different time periods. In practice, you encounter dependent samples when you have:
  • Paired testing like before-and-after experiments.
  • Repeated measurements on the same subjects.
  • Matched samples based on certain criteria (like age, gender, etc.).
The most important aspect is that each sample in the pair is not independently chosen; there鈥檚 a logical linkage between them. For example, if a group of patients' health metrics are measured before and after a treatment, these represent dependent samples. The paired nature of the study ensures that the influence of initial conditions is minimized when evaluating the treatment effect.
Independent Samples
Independent samples are those where the selection of one sample doesn't influence or alter the selection of another. They are the cornerstone of many statistical hypotheses tests where comparisons need to be made. In more technical terms:
  • The two samples are chosen completely at random.
  • Changes in one sample do not impact the other sample.
  • There is no imposed connection between the samples.
For instance, comparing the test scores of two different schools would require independent samples, as the students in one school don't affect the students in the other. This type of sampling is prevalent since it eliminates bias that can stem from sample selection. Thus, it allows for more valid generalizations across different groups or methods.
Statistical Methods Comparison
When comparing statistical methods, one often needs to decide whether the data consists of dependent or independent samples. This decision determines which statistical test should be employed. Here are some key points to consider during comparison:
  • Choice of Test: Dependent samples often use paired tests, such as a paired t-test. For independent samples, you'll likely utilize tests like an independent t-test or ANOVA.
  • Assumptions: Understand the assumptions underlying each test, such as the need for normal distribution or equal variances.
  • Impact: Knowing the relationship between samples can significantly influence the interpretation and outcome of the results.
The chemist's problem provides a clear example. Since the measurements from the proposed method and the current method aren't paired but taken independently, you'd apply an independent sample comparison. This removes the doubt that results are affected by any inherent relation, ensuring the evaluation process purely assesses the method's efficacy.

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Most popular questions from this chapter

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Lauren, a brunette, was tired of hearing, "Blondes have more fun." She set out to "prove" that "brunettes are more intelligent." Lauren randomly (as best she could) selected 40 blondes and 40 brunettes at her high school.$$\begin{array}{llll}\hline \text { Blondes } & n_{\beta}=40 & \bar{x}_{\beta i}=88.375 & s_{\beta i}=6.134\\\\\text { Brunettes } & n_{\mathrm{S}}=40 & \bar{x}_{A r}=87.600 & s_{B r}=6.640\\\\\hline\end{array}$$ Upon seeing the sample results, does Lauren have support for her claim that "brunettes are more intelligent than blondes"? Explain. What could Lauren say about blondes' and brunettes' intelligence?
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