91Ó°ÊÓ

It is known that roughly \(2 / 3\) of all human beings have a dominant right foot or eye. Is there also right-sided dominance in kissing behavior? The article "Human Behavior: Adult Persistence of Head-Turning Asymmetry" (Nature, 2003: 771) reported that in a random sample of 124 kissing couples, both people in 80 of the couples tended to lean more to the right than to the left. a. If \(2 / 3\) of all kissing couples exhibit this right-leaning behavior, what is the probability that the number in a sample of 124 who do so differs from the expected value by at least as much as what was actually observed? b. Does the result of the experiment suggest that the \(2 / 3\) figure is implausible for kissing behavior? State and test the appropriate hypotheses.

Step by step solution

01

Define the Hypotheses

We need to test if the observed proportion of right-leaning couples differs from the hypothesized proportion of \( \frac{2}{3} \). The null hypothesis \( H_0 \) is that the proportion of right-leaning couples is \( \frac{2}{3} \). The alternative hypothesis \( H_a \) is that the proportion is different from \( \frac{2}{3} \), i.e., \( H_0: p = \frac{2}{3} \) versus \( H_a: p eq \frac{2}{3} \).

02

Calculate the Test Statistic

First, we calculate the sample proportion of right-leaning couples, \( \hat{p} = \frac{80}{124} \approx 0.6452 \). The test statistic is calculated using the formula for a proportion: \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \), where \( n \) is the sample size, and \( p_0 = \frac{2}{3} \).Substitute the values: \[ z = \frac{0.6452 - \frac{2}{3}}{\sqrt{\frac{\frac{2}{3}(\frac{1}{3})}{124}}} = \frac{0.6452 - 0.6667}{0.0429} \approx -0.5011 \]

03

Determine the P-Value

Since this is a two-tailed test, we find the probability that the absolute value of the test statistic is greater than \( 0.5011 \). Using a standard normal distribution table or calculator, the p-value associated with a \( z \)-score of \( -0.5011 \) is roughly 0.615. Thus, the p-value for the two-tailed test is \( 2 \times 0.308 = 0.616 \).

04

Make a Decision

Compare the p-value to the significance level \( \alpha = 0.05 \). Since \( 0.616 > 0.05 \), we fail to reject the null hypothesis \( H_0 \). This means that we do not have sufficient evidence to suggest that the proportion of right-leaning couples differs from \( \frac{2}{3} \).

05

Conclusion

The calculated p-value suggests that there is not enough statistical evidence to conclude that the population proportion of right-leaning kissing couples is different from \( \frac{2}{3} \). Therefore, the original figure hypothesized remains plausible given the sample data.

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Key Concepts

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

Proportion Test

Proportion tests are a type of hypothesis test used when we're dealing with proportions rather than means. In this particular exercise, we're interested in whether the proportion of right-leaning kissing couples differs from a hypothesized proportion of \( \frac{2}{3} \). Proportion tests help determine if the observed proportion \( \hat{p} \) deviates significantly from the expected proportion \( p_0 \). This is particularly useful in studies where you compare a sample statistic to a known value or to another population proportion. To conduct a proportion test, follow these steps:

• Determine your null and alternative hypotheses based on the proportions you are comparing.
• Calculate the test statistic, which is typically a z-score for proportions.
• Use the test statistic to find the p-value, which indicates the probability of observing the data given the null hypothesis is true.
• Compare the p-value to a significance level (usually \( \alpha = 0.05 \)) to determine if you should reject or fail to reject the null hypothesis.

Null Hypothesis

In hypothesis testing, the null hypothesis, denoted as \( H_0 \), is the statement being tested. In this exercise, the null hypothesis asserts that the proportion of right-leaning kissing couples is \( \frac{2}{3} \). This figure assumes that kissing behavior follows the same right-sided dominance seen in other human actions. The null hypothesis is typically the status quo or a statement of no effect. It serves as a foundation for testing the opposite claim, known as the alternative hypothesis. By starting with the null hypothesis, researchers look for evidence against it to see if there's enough support for a new claim.Key points about the null hypothesis include:

• It is usually a statement of equality, like \( p = 0.667 \).
• It's an assumption that there's no difference or effect.
• The goal is to test whether observed data significantly contradict this assumption.

Alternative Hypothesis

The alternative hypothesis,\( H_a \), opposes the null hypothesis. In our analysis, the alternative hypothesis suggests that the proportion of right-leaning kissing couples is not \( \frac{2}{3} \). This alternative hypothesis addresses whether there is any deviation from the right - leaning behavior, regardless of the direction of the difference. In hypothesis testing, the alternative hypothesis is what you hope to support with evidence. Important aspects about the alternative hypothesis:

• Its form depends on the research question and could be two-sided (as in this problem where \( H_a: p eq \frac{2}{3} \)) or one-sided.
• It represents a new effect, new difference, or change the study aims to detect.
• It guides the conclusion of whether there's a statistically significant effect based on the data analysis.

P-Value Interpretation

The p-value is a critical aspect of hypothesis testing because it helps you decide whether to reject the null hypothesis. It represents the probability of observing test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is true. For this exercise, a p-value of 0.616 was calculated. This tells us how likely the collected data would occur if the true proportion of right-leaning kissing couples actually was \( \frac{2}{3} \). Understanding p-value significance:

• A high p-value (greater than \( \alpha = 0.05 \)) suggests the data is consistent with the null hypothesis. Here, failing to reject \( H_0 \) indicates you do not have enough evidence of a proportion different than \( \frac{2}{3} \).
• A low p-value (less than \( \alpha \)) provides evidence against the null hypothesis, suggesting a significant deviation from the expected proportion.
• The usual threshold, \( \alpha = 0.05 \), is a common level of significance determining when to consider the results statistically significant.