To test H0: μ = 100 versus H1: μ ≠ 100, a simple random sample of size n = 23 is obtained from a population that is known to be normally distributed.


d. Will the researcher reject the null hypothesis? Why?

Verified step by step guidance

1. Identify the null hypothesis \(H_0: \mu = 100\) and the alternative hypothesis \(H_1: \mu \neq 100\), which indicates a two-tailed test.
2. Determine the sample size \(n = 23\) and note that the population is normally distributed, so the \(t\)-distribution will be used because the population standard deviation is unknown.
3. Calculate the test statistic using the formula \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(\mu_0 = 100\) is the hypothesized mean, and \(s\) is the sample standard deviation.
4. Find the critical \(t\)-value(s) from the \(t\)-distribution table for a two-tailed test with degrees of freedom \(df = n - 1 = 22\) at the chosen significance level (commonly \(\alpha = 0.05\)).
5. Compare the calculated test statistic to the critical \(t\)-value(s): if the absolute value of the test statistic is greater than the critical value, reject the null hypothesis; otherwise, do not reject it.

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 (H1). It involves calculating a test statistic from sample data and comparing it to a critical value or using a p-value to make a decision.

t-Distribution and Degrees of Freedom

When the population standard deviation is unknown and the sample size is small (n < 30), the t-distribution is used instead of the normal distribution. The shape of the t-distribution depends on degrees of freedom, calculated as n - 1, which affects critical values for hypothesis testing.

Decision Rule for Rejecting the Null Hypothesis

The decision to reject H0 depends on whether the test statistic falls into the rejection region defined by the significance level (α). For a two-tailed test, if the absolute value of the test statistic exceeds the critical t-value, or if the p-value is less than α, the null hypothesis is rejected.