91Ó°ÊÓ

Vigorous exercise helps people live several years longer (on average). Whether mild activities like slow walking extend life is not clear. Suppose that the added life expectancy from regular slow walking is just 2 months. A statistical test is more likely to find a significant increase in mean life expectancy if (a) it is based on a very large random sample and a \(5 \%\) significance level is used. (b) it is based on a very large random sample and a \(1 \%\) significance level is used. (c) it is based on a very small random sample and a \(5 \%\) significance level is used. (d) it is based on a very small random sample and a \(1 \%\) significance level is used. (e) the size of the sample doesn't have any effect on the significance of the test.

Step by step solution

01

Understand the Effect Size

Here, the added life expectancy from regular slow walking is just 2 months. This is a small effect size, which means detecting this small difference using statistical tests requires certain conditions.

02

Importance of Sample Size

Larger sample sizes tend to produce more precise estimates of a population parameter. Thus, a large sample size is better for detecting small differences, like a 2-month increase in life expectancy.

03

Understand Significance Levels

The significance level (e.g., 5% or 1%) influences the probability of rejecting the null hypothesis when it is actually true (Type I error). A 5% significance level means there's a 5% chance of a Type I error, and a 1% level is more stringent.

04

Evaluate Options Based on Sample Size and Significance Level

For small effect sizes, a larger sample and a less stringent significance level (such as 5%) will provide a test that is more likely to detect this effect, assuming it exists.

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Key Concepts

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

Sample Size

The sample size in a statistical study refers to the number of observations or data points collected. This is a crucial factor because the larger your sample size, the closer your sample mean is likely to be to the true population mean. When you're trying to detect small differences, such as a 2-month increase in life expectancy from slow walking, having a large sample size is particularly important.
Here’s why:

• Larger sample sizes reduce variability, increasing the precision of your estimates.
• With more data, the distribution of sample means tightens up around the true population mean, making it easier to detect even small changes.

Think of it like tuning in a radio station; the more precise your tuning, the clearer the signal you receive. Similarly, a larger sample size "tunes" in the signal of small effects more clearly.

Effect Size

Effect size is a quantitative measure of the magnitude of the experimental effect. In our example, it's the 2-month increase in life expectancy due to slow walking. The size of this effect helps determine how easy or hard it will be to notice any differences in your data.
Here are some key points about effect size:

• Small effect sizes require larger samples to be detected reliably, as there is less "contrast" against the natural variability.
• Understanding effect size helps in designing more effective studies and choosing the appropriate statistical tests.

The smaller the effect size, the more difficult it is to distinguish from random noise in the data. This is why large samples are often necessary for detecting small effects in statistical testing.

Significance Level

The significance level, often denoted by alpha (α), is the probability threshold for rejecting the null hypothesis. Common levels include 5% (0.05) or 1% (0.01). It essentially tells us how willing we are to risk a Type I error, which is rejecting a true null hypothesis.
Here's how significance levels affect your test:

• A 5% significance level means you accept a 5% risk of concluding that a difference exists when, in fact, it doesn't.
• A 1% significance level lowers this risk, making your test more stringent, but potentially missing true effects if they are small.

In scenarios with small effect sizes, like our slow walking example, using a 5% significance level with a large sample increases the likelihood of detecting a real difference, albeit with a slightly higher risk of a Type I error.

Type I Error

A Type I error occurs when the null hypothesis is true, but we incorrectly reject it. It’s like a false alarm, where we think there's an effect, but there's none. The probability of making a Type I error is determined by the significance level you choose for your test. Understanding Type I error involves:

• Accepting that with a 5% significance level, 1 out of every 20 tests might indicate an effect that isn't there.
• Balancing the risks of Type I errors with the ability to detect actual effects, especially when dealing with small effect sizes.

Choosing the right balance of sample size, effect size, and significance level is crucial. This helps ensure that you're able to distinguish between true signals in the data and noise, avoiding unnecessary Type I errors.