A simple random sample of size n = 19 is drawn from a population that is normally distributed. The sample mean is found to be 0.8, and the sample standard deviation is found to be 0.4. Test whether the population mean is less than 1.0 at the α = 0.01 level of significance.

Verified step by step guidance

1. Identify the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). Since we are testing if the population mean is less than 1.0, set \(H_0: \mu = 1.0\) and \(H_a: \mu < 1.0\).
2. Determine the test statistic to use. Because the population standard deviation is unknown and the sample size is small (\(n=19\)), use the t-test statistic, which is calculated as: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where \(\bar{x}\) is the sample mean, \(\mu_0\) is the hypothesized population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.
3. Calculate the degrees of freedom for the t-distribution, which is \(df = n - 1\).
4. Find the critical value from the t-distribution table corresponding to \(\alpha = 0.01\) for a one-tailed test with \(df\) degrees of freedom. This critical value defines the rejection region for the test.
5. Compare the calculated t-statistic to the critical value: if the t-statistic is less than the negative of the critical value (since this is a left-tailed test), 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 about a population parameter. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine if H0 can be rejected at a given significance level (α).

t-Distribution and t-Test

The t-distribution is used instead of the normal distribution when the population standard deviation is unknown and the sample size is small (n < 30). The one-sample t-test compares the sample mean to a hypothesized population mean, accounting for sample variability using the sample standard deviation.

Significance Level and Critical Value

The significance level (α) is the probability of rejecting the null hypothesis when it is true (Type I error). For a given α, a critical value from the t-distribution defines the rejection region. If the test statistic falls into this region, the null hypothesis is rejected in favor of the alternative.