To test H0: mu = 100 versus Ha: mu > 100, a simple random sample of size n = 35 is obtained from an unknown distribution. The sample mean is 104.3 and the sample standard deviation is 12.4.
b. Use the classical or p-value approach to decide whether to reject the statement in the null hypothesis at the alpha = 0.05 level of significance.

Verified step by step guidance

1. Identify the null hypothesis \(H_0: \mu = 100\) and the alternative hypothesis \(H_a: \mu > 100\). This is a right-tailed test because the alternative hypothesis is looking for values greater than 100.
2. Calculate the test statistic using the formula for the one-sample t-test since the population standard deviation is unknown and the sample size is less than 30 or the distribution is unknown: \( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \) where \(\bar{x} = 104.3\) is the sample mean, \(\mu_0 = 100\) is the hypothesized population mean, \(s = 12.4\) is the sample standard deviation, and \(n = 35\) is the sample size.
3. Determine the degrees of freedom for the t-distribution, which is \(df = n - 1 = 35 - 1 = 34\).
4. Using the calculated t-statistic and degrees of freedom, find the p-value from the t-distribution. The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.
5. Compare the p-value to the significance level \(\alpha = 0.05\). If the p-value is less than \(\alpha\), reject the null hypothesis \(H_0\); otherwise, do not reject \(H_0\). This decision tells you whether there is enough evidence to support the claim that \(\mu > 100\).

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis (H0) in favor of an alternative hypothesis (Ha). It involves setting a significance level (alpha), calculating a test statistic from sample data, and comparing it to a critical value or using a p-value to make a decision.

t-Test for a Single Mean

When the population standard deviation is unknown and the sample size is relatively small, the t-test is used to test hypotheses about the population mean. The test statistic follows a t-distribution with n-1 degrees of freedom, calculated using the sample mean, sample standard deviation, and sample size.

p-Value and Significance Level

The p-value measures the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. If the p-value is less than the chosen significance level (alpha), the null hypothesis is rejected, indicating sufficient evidence to support the alternative hypothesis.