91Ó°ÊÓ

A psychological experiment was done to investigate the effect of anxiety on a person's desire to be alone or in company (Schacter \(1959 ;\) Lehmann 1975 ). A group of 30 subjects was randomly divided into two groups of sizes 13 and \(17 .\) The subjects were told that they would be subjected to some electric shocks, but one group was told that the shocks would be quite painful and the other group was told that they would be mild and painless. The former group was the "high-anxiety" group, and the latter was the "low-anxiety" group. Both groups were told that there would be a 10 -min wait before the experiment began, and each subject was given the choice of waiting alone or with the other subjects. The following are the results: $$\begin{array}{c|c|c} \hline & \text { Wait Together } & \text { Wait Alone } \\ \hline \text { High-Anxiety } & 12 & 5 \\ \hline \text { Low-Anxiety } & 4 & 9 \end{array}$$ Use Fisher's exact test to test whether there is a significant difference between the high- and low-anxiety groups.

Step by step solution

01

Understand the Hypotheses

In Fisher's exact test, we start by defining the null and alternative hypotheses. Here, the null hypothesis \((H_0)\) is that there is no association between anxiety level and the preference to wait alone or together. The alternative hypothesis \((H_a)\) is that there is an association between anxiety level and the waiting preference.

02

Set Up the Contingency Table

Organize the data into a 2x2 contingency table:\[\begin{array}{c|c|c}\hline & \text{Wait Together} & \text{Wait Alone} \\hline \text{High-Anxiety} & 12 & 1 \\hline \text{Low-Anxiety} & 4 & 9 \\end{array}\]

03

Calculate the Test Statistic

The test statistic for Fisher's exact test is the probability of observing the data given the null hypothesis. The formula is:\[P = \frac{{\binom{12}{12} \binom{13-12}{5} \binom{4}{4} \binom{17-4}{9}}}{{\binom{30}{13}}}\]These binomial coefficients represent different ways to choose subjects for each outcome.

04

Interpret the Result

Compute the probabilities and evaluate whether they indicate statistical significance. Fisher's exact test output is typically a p-value. For this table:- Calculate the exact probability using the hypergeometric distribution.- Compare the p-value to a significance level (e.g., \( \alpha = 0.05 \)).- A p-value less than \( \alpha \) indicates a significant association.

05

Make a Conclusion

Based on the p-value: - If the p-value < 0.05, reject the null hypothesis, implying there is a significant difference in preferences between high-and low-anxiety groups. - If the p-value >= 0.05, do not reject the null hypothesis; no significant difference in preferences is observed.

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Key Concepts

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

Contingency Table

A contingency table is a useful way to organize data that shows the frequency distribution of variables. It's often used in statistics to reveal the relationship between two variables. In the context of the anxiety experiment described, the contingency table has two rows and two columns, representing two different anxiety groups and their preference of waiting settings: together or alone.
The rows represent the different levels of anxiety: high and low. The columns show the choice of waiting alone or with others. By arranging the data in this way, it's easier to apply statistical tests, like Fisher's exact test, to determine if there's a relationship between these variables.
For instance, such a table allows us to quickly see that 12 people in the high-anxiety group preferred to wait together, while 9 from the low-anxiety group preferred to wait alone. It's a concise method to view the data and helps in performing further statistical analyses.

Null Hypothesis

In statistical testing, the null hypothesis is a statement that there is no effect or no association between variables. It's what researchers aim to test against to find statistical evidence. For the anxiety experiment, the null hypothesis (denoted as \(H_0\)) states that there’s no difference between the high-anxiety and low-anxiety groups in terms of their preference to wait alone or together.
Essentially, the null hypothesis suggests that any differences observed in the data occur purely by chance and not due to the experimental condition. A statistical test like Fisher's exact test is then used to determine if we have enough evidence to reject this statement. If we find a significant result, we have grounds to believe there is a real effect or difference, but if not, we retain the null hypothesis.

P-Value

The p-value is a measure used in statistics to assess the strength of evidence against the null hypothesis. It quantifies the probability of observing data as extreme as, or more extreme than, the observed results if the null hypothesis were true. In the anxiety experiment, Fisher’s exact test calculates a p-value to determine if the observed preference differences are statistically significant.
A small p-value, typically less than a significance level of 0.05, indicates strong evidence against the null hypothesis. This means it's unlikely the observed differences happened by chance, suggesting a real effect or association. Conversely, a higher p-value suggests that the observed differences could have occurred by random chance, and there is not enough evidence to reject the null hypothesis.

Statistical Significance

Statistical significance is a determination that a relationship between variables in a test is not likely due to chance. It helps researchers understand whether their results are meaningful beyond natural variability. In our psychological experiment example, we determine statistical significance by comparing the p-value to a predefined threshold, often 0.05.
The concept of statistical significance doesn't comment on the magnitude or practical importance of the difference, only the likelihood that it could be attributed to chance instead of a real effect. When a p-value is below this threshold, we say the results are statistically significant, suggesting that anxiety level does indeed influence the choice to wait alone or with others.
If the results were statistically significant, this would imply a noticeable difference between the preferences of the high- and low-anxiety groups, validating the alternative hypothesis of an association between anxiety level and waiting preference.