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Chapter 18: Problem 24

Vigorous exercise is associated with several years' longer life (on the average). Researchers in Denmark found evidence that slow jogging may provide even better life- expectancy benefits than more vigorous running. \({ }^{8}\) Suppose that the added life expectancy associated with slow jogging steadily for 30 minutes three times a week is just one month. A statistical test is more likely to find a significant increase in mean life expectancy for those who jog slowly if (a) it is based on a very large random sample. (b) it is based on a very small random sample. (c) The size of the sample has little effect on significance for such a small increase in life expectancy.

Short Answer

Expert verified
(a) it is based on a very large random sample.

Step by step solution

01

Understanding the Question

We need to determine which condition (a large or small sample size) makes it more likely for a statistical test to find a significant effect of slow jogging on increased life expectancy.
02

Interpreting Statistical Significance

A statistical test is used to determine whether an observed effect is likely to have occurred by chance. The larger the sample size, the more power the test has to detect significant effects, even if they are small.
03

Analyzing Effect of Sample Size

A very large random sample increases the statistical power of a test. This means it is more sensitive to detect small differences or effects, like a small increase in life expectancy due to slow jogging.
04

Concluding with an Answer

Based on statistical principles, if the effect size is small, such as a one-month increase in life expectancy, a larger sample size increases the likelihood of detecting a significant effect. Thus, option (a) is more likely to lead to a significant result.

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Key Concepts

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

Sample Size
In statistical research, the term 'sample size' refers to the number of observations or data points collected from a larger population. Choosing the right sample size is crucial because it influences the reliability and accuracy of study results. When you're analyzing something like life expectancy, increasing the sample size helps you to draw more precise conclusions.

A larger sample size can offer several benefits:
  • Reduced margin of error: With more data, the variability decreases, leading to more accurate estimates.
  • Increased confidence in results: Larger samples provide more reliable insights that are less likely to be due to random chance.
  • Enhanced statistical power: When it comes to detecting small effects, such as a one-month increase in life expectancy, having more data improves your ability to discern these changes.
Thus, when researchers aim to study the effects of slow jogging on life expectancy, a larger sample size offers more trustworthy and actionable findings.
Life Expectancy
Life expectancy is a measure of the average number of years remaining at a given age, reflecting overall quality and length of life throughout a population. Lifestyle choices, such as diet and exercise, can have significant impacts on life expectancy. In the exercise, the notion was to understand how a regular activity like slow jogging could influence this measure.

Key Points about Life Expectancy:
  • It encompasses various health factors: Genetics, environment, and personal habits can all contribute to how long we live.
  • Small increases are meaningful: Even a one-month increase, as mentioned in the problem, can translate into a critical difference when scaled to a large population.
  • Comparisons provide insights: By comparing life expectancy in joggers versus non-joggers, researchers can infer the potential benefits of physical activity.
Understanding how factors like exercise affect our longevity helps frame public health recommendations and personal lifestyle decisions.
Statistical Power
Statistical power is a measure of a test's ability to correctly reject a false null hypothesis. In simple terms, it reflects the test's capacity to detect true effects that exist in the data.

When it comes to studying life expectancy, statistical power becomes crucial for the following reasons:
  • Detecting small effects: As in the exercise scenario, the power to detect a small increase in life expectancy, such as one from slow jogging, hinges on having enough samples to confirm the change isn't due to random chance.
  • Sample size connection: A higher sample size typically enhances statistical power, making it easier to identify even minimal effects.
  • Avoiding Type II errors: With sufficient power, researchers are less likely to miss an existing effect, which is key when assessing health interventions.
Therefore, by focusing on increasing the sample size, you maximize the test's power, ensuring that any observed impact of activities like jogging on life expectancy isn't overlooked but validated through robust research design.

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Most popular questions from this chapter

You read that a statistical test at the \(\alpha=0.05\) level has probability \(0.49\) of making a Type II error when a specific alternative is true. What is the power of the test against this alternative?

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