[DATA] Bull and Bear Markets The stock market goes through periods known as bull markets and bear markets. A bull market exists when the stock market increases in value by at least 20%, while a bear market exists when the stock market decreases in value by at least 20%. The data below represent the number of months a random sample of bull and bear markets have lasted.

. Although both distributions are skewed, it is likely reasonable to invoke the Central Limit Theorem and conclude the sample mean difference is approximately normal due to the fact the sample size is 25 for each sample. Plus, the two-sample t-test is robust so minor departures from the normality condition are acceptable. In light of this, perform a two-sample t-test on the data. What is the P-value for this test?

Verified step by step guidance

1. Identify the hypotheses for the two-sample t-test. The null hypothesis \(H_0\) states that the population means are equal: \(\mu_1 = \mu_2\). The alternative hypothesis \(H_a\) states that the population means are not equal: \(\mu_1 \neq \mu_2\).
2. Calculate the standard error (SE) of the difference between the two sample means using the formula: \(SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\) where \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.
3. Compute the test statistic (t) using the formula: \(t = \frac{\bar{x}_1 - \bar{x}_2}{SE}\) where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
4. Determine the degrees of freedom (df) for the two-sample t-test. Since the sample sizes and variances are different, use the Welch-Satterthwaite approximation: \(df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}\)
5. Use the calculated t statistic and degrees of freedom to find the P-value from the t-distribution. This P-value indicates the probability of observing a difference as extreme as the one in the sample data if the null hypothesis is true.

Key Concepts

Two-Sample t-Test

A two-sample t-test compares the means of two independent groups to determine if there is a statistically significant difference between them. It uses sample means, standard deviations, and sample sizes to calculate a test statistic, which is then used to find the P-value. This test assumes the samples are independent and approximately normally distributed or that the sample size is large enough for the Central Limit Theorem to apply.

Central Limit Theorem (CLT)

The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution shape. This allows the use of normal-based inference methods, like the t-test, even when the original data are skewed, provided the sample size is sufficiently large (usually n ≥ 25).

P-value Interpretation

The P-value measures the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true. A small P-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting a significant difference between the groups.