91Ó°ÊÓ

In one segment of the TV series MythBusters, an experiment was conducted to test the common belief that people are more likely to yawn when they see others yawning. In one group, 34 subjects were exposed to yawning, and 10 of them yawned. In another group, 16 subjects were not exposed to yawning, and 4 of them yawned. We want to test the belief that people are more likely to yawn when they are exposed to yawning. a. Why can't we test the claim using the methods of this section? b. If we ignore the requirements and use the methods of this section, what is the \(P\) -value? How does it compare to the \(P\) -value of \(0.5128\) that would be obtained by using Fisher's exact test? c. Comment on the conclusion of the Mythbusters segment that yawning is contagious.

Step by step solution

01

Identify Statistical Methods

The problem involves comparing the proportions of yawning between two groups. The standard methods discussed in this section may not be appropriate because the sample sizes are small and the conditions for the normal approximation to the binomial distribution might not be satisfied.

02

Calculate Proportions

Calculate the proportion of people who yawned in each group.Group exposed to yawning: \[ \text{Proportion} = \frac{10}{34} = 0.294 \]Group not exposed to yawning:\[ \text{Proportion} = \frac{4}{16} = 0.25 \]

03

State the Hypotheses

State the null and alternative hypotheses.\[ H_0: p_1 - p_2 = 0 \]\[ H_a: p_1 - p_2 > 0 \]where \( p_1 \) is the proportion of yawning in the exposed group, and \( p_2 \) is the proportion in the non-exposed group.

04

Calculate the Test Statistic

Use the formula for the test statistic for comparing two proportions:\[ z = \frac{(\frac{10}{34} - \frac{4}{16})}{\text{Standard Error}} \]The standard error (SE) is given by:\[ SE = \frac{10+4}{34+16} \times \frac{1}{34} + \frac{1}{16} \]Calculate SE and then the \( z \)-value.

05

Find the P-Value

Determine the p-value using the calculated z-value. Compare it with the \( p \)-value given by Fisher's exact test.

06

Compare P-Values and Make a Conclusion

The p-value using our method should be compared with the p-value from Fisher's exact test (0.5128). If our obtained p-value is larger, it's an indication that our method is less accurate for this test. Since our method might not be suitable due to sample sizes and conditions, Fisher's exact test is more reliable.

07

Comment on Mythbusters' Conclusion

Based on the Fisher's exact test p-value (0.5128), which is greater than 0.05, we fail to reject the null hypothesis. Thus, the conclusion that yawning is contagious is not statistically supported by the data.

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Key Concepts

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

Comparative Statistics

In statistics, comparative studies are critical when analyzing differences between two or more groups. In our context, the experiment aimed to compare the proportion of people who yawned when exposed to yawning against those who were not exposed. Comparative statistics help establish if the observed differences are due to chance or a significant factor.

There are numerous methods to perform such comparisons, each suited to different conditions:

• Parametric tests like t-tests for means comparison when data is normally distributed.
• Non-parametric tests like Mann-Whitney U when data is not normally distributed.
• For proportions, which is our current interest, z-tests or Chi-square tests are commonly used.

The chosen method must align with the sample size and data distribution. When these conditions are not met, alternative methods such as Fisher's exact test can be more appropriate.

Proportion Comparison

Proportion comparison is about analyzing if the proportions of a particular occurrence are significantly different between two groups. For this exercise, we compared the proportions of people who yawned:

• In the exposed group: \(\frac{10}{34} \)\text {which equals 0.294}
• In the non-exposed group: \(\frac{4}{16} \)\text {which equals 0.25}

To test for differences, we establish hypotheses:

• The null hypothesis \((H_0)\): The proportions are equal, implying that yawning is not more frequent in the exposed group.
• Alternative hypothesis \((H_a)\): The proportion in the exposed group is larger, meaning yawning is contagious.

We then calculate the test statistic (z-value), where a high value would suggest a significant difference between the groups.

Fisher's Exact Test

Fisher's exact test is used to determine the significance of the association between two kinds of classifications. It’s particularly useful when sample sizes are small and the conditions for normal approximation are not met.

In our yawning experiment, since the sample sizes are small (34 and 16), Fisher's exact test is a better alternative:

• It directly computes the exact probability of obtaining the observed distribution.
• Avoids the inaccuracies of large-sample z-tests or chi-square tests.

Using Fisher's exact test for our data, we got a p-value of 0.5128, showing no significant difference in the proportions of yawning between the two groups.

P-Value Interpretation

The p-value is a measure that helps you determine the strength of your evidence against the null hypothesis:

• If the p-value is less than a chosen significance level (often 0.05), there is strong evidence against the null hypothesis, implying a significant difference.
• If the p-value is greater, we do not have enough evidence to reject the null hypothesis.

In our scenario, using Fisher's exact test, the p-value is 0.5128, which is greater than 0.05, suggesting that:

• We fail to reject the null hypothesis.
• There is no statistically significant evidence to support the claim that yawning is more frequent when exposed to others yawning.

Therefore, despite the Mythbusters’ segment, according to this statistical test, we cannot conclusively state yawning is contagious.