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Chapter 9: Problem 55

Fishing for significance A marketing study conducts 60 significance tests about means and proportions for several groups. Of them, 3 tests are statistically significant at the 0.05 level. The study's final report stresses only the tests with significant results, not mentioning the other 57 tests. What is misleading about this?

Short Answer

Expert verified
The report is misleading due to possible false positives not considered.

Step by step solution

01

Understanding the Alpha Level

The significance level of 0.05 means there's a 5% chance of finding a statistically significant result by random chance, even if there's no real effect or difference.
02

Calculating Expected False Positives

With 60 significance tests conducted at a 0.05 significance level, the expected number of false positives (incorrectly identified as significant) is calculated as: \( 60 \times 0.05 = 3 \). This implies that about 3 of the tests could be significant just by random chance.
03

Analyzing the Results

Since exactly 3 tests were found to be significant and this matches the expected number of false positives, it's likely that these results occurred by chance rather than reflecting true differences or effects.
04

Identifying What is Misleading

The report is misleading because it highlights only the 3 significant results without accounting for the probability of those being false positives. This selection bias can lead to incorrect conclusions since the significance of these results can be attributed to random chance.

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Key Concepts

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

Multiple Testing
When conducting multiple statistical tests, the probability of finding at least one statistically significant result by random chance increases. Think of it like flipping a coin 60 times and expecting at least a few heads. Each test has its own probability of a false positive, and when you do multiple tests, those probabilities add up.
You can consider each test as an independent event, much like tossing a coin. If one test has a 5% chance of showing a false positive result, running many tests multiplies this likelihood of finding spurious associations.
  • This is why researchers prefer to adjust the significance level or use techniques like Bonferroni correction when many tests are involved.
  • This helps control the overall error rate, so you don't end up drawing conclusions based on what could just be random noise.
To maintain the integrity of the results, be mindful of this testing multiplication risk.
False Positive Rate
The false positive rate is an inherent part of statistical testing. It's the probability of detecting a supposed effect or difference when in reality none exists. Consider it a false alarm in your test results.
  • The rate is usually set by researchers through the significance level, often denoted by alpha (伪).
  • If 伪 is 0.05, you expect about 5% of your findings to be false positives if there are no true effects present.
When multiple tests are conducted, the cumulative false positive rate increases, leading to a greater likelihood of seeing at least one false positive result. This is why it's essential to properly account for these rates when interpreting results so that you don't overestimate the significance of your findings.
Alpha Level
The alpha level, commonly set at 0.05, represents a threshold in hypothesis testing. It's the maximum probability of wrongly rejecting a true null hypothesis, essentially marking how often researchers are willing to be wrong.
  • A 0.05 alpha means there's a 5% risk that the test will identify a false positive as significant.
  • It's a critical part of the testing process used to determine whether results are considered statistically significant.
If one does many tests, as in the original exercise, the number of expected false positives rises. Thus, the alpha level should be carefully selected in contexts with extensive testing to avoid misleading conclusions.
Selection Bias
Selection bias occurs when the data collected, or results emphasized, do not represent the entire set or possibility space. This could lead to interpretations that are skewed or not reflective of reality.
In the context of the exercise, focusing only on tests that show statistically significant results (like the three out of sixty tests) without considering the rest presents a biased view.
  • This can occur because these tests may have shown significance due to chance rather than an actual effect.
  • Overemphasizing such results without mentioning the possibility of false positives can mislead others into believing non-existent effects.
To prevent selection bias, all results, including non-significant ones, should be considered in the analysis and interpretation phase.

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Most popular questions from this chapter

P-value Indicate whether each of the following P-values gives strong evidence or not especially strong evidence against the null hypothesis. a. 0.38 b. 0.001

\(\mathbf{H}_{0}\) or \(\mathbf{H}_{a}\) ? For parts a and \(\mathrm{b},\) is the statement a null hypothesis, or an alternative hypothesis? a. In Canada, the proportion of adults who favor legalized gambling equals \(0.50 .\) b. The proportion of all Canadian college students who are regular smokers is less than 0.24 , the value it was 10 years ago. c. Introducing notation for a parameter, state the hypotheses in parts a and \(\mathrm{b}\) in terms of the parameter values.

Gender bias in selecting managers Exercise 9.19 tested the claim that female employees were passed over for management training in favor of their male colleagues. Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training is \(40 \%\) female and \(60 \%\) male. Let \(p\) be the probability of selecting a female for any given selection. For testing \(\mathrm{H}_{0}: p=0.40\) against \(\mathrm{H}_{a}: p<0.40\) based on a random sample of 50 selections, using the 0.05 significance level, verify that: a. A Type II error occurs if the sample proportion falls less than 1.645 standard errors below the null hypothesis value, which means that \(\hat{p}>0.286\). b. When \(p=0.20,\) a Type II error has probability 0.06 .

Error probability A significance test about a proportion is conducted using a significance level of \(0.05 .\) The test statistic equals \(2.58 .\) The P-value is 0.01 a. If \(\mathrm{H}_{0}\) were true, for what probability of a Type I error was the test designed? b. If this test resulted in a decision error, what type of error was it?

Errors in the courtroom Consider the test of \(\mathrm{H}_{0}\) : The defendant is not guilty against \(\mathrm{H}_{a}:\) The defendant is guilty. a. Explain in context the conclusion of the test if \(\mathrm{H}_{0}\) is rejected. b. Describe the consequence of a Type I error. c. Explain in context the conclusion of the test if you fail to reject \(\mathrm{H}_{0}\) d. Describe the consequence of a Type II error.
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