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A test statistic helps us to determine if our sample data significantly deviates from the null hypothesis. It's a standardized value that tells us how many standard deviations our sample mean is away from the population mean under the assumption that the null hypothesis is true. In our problem, the test statistic is calculated using the formula: \ \[ z = \frac{\bar{x} - u}{\frac{u}{u}} \] We use the given sample mean (\bar{x} = 47.1), population mean (\bar{x} = 50), standard deviation (\bar{x} = 12), and sample size ( = 24). By plugging in these values, we get \ \( z \ = \ \bar\bar{x} - 50 = \qquad {-2.9}{2.45} = -1.18 \). This means our sample mean is \ approximately \ -1.18 standard deviations below the population mean.

The critical value is a threshold that our test statistic must exceed to reject the null hypothesis. It depends on the significance level ( Δ ħ e) of the test. A common significance level used is 0.05, meaning there's a 5% risk of rejecting the null hypothesis when it is actually true. In a one-tailed test, we focus on one end of the distribution. For our problem, we look at the left tail because the alternative hypothesis suggests fewer than 50. Using the z-table, we find the z-score corresponding to 0.05, which is approximately -1.645. This score marks the boundary below which we reject the null hypothesis.

A normal distribution is a bell-shaped curve that is symmetric about the mean. In hypothesis testing, we assume the population from which the sample is drawn follows a normal distribution. This allows us to use z-scores to determine probabilities and critical values. In our situation, we use the normal curve to visualize regions of acceptance and rejection based on standard deviations from the mean. When we shade the area to the left of the critical value -1.645, we define the critical region. If our test statistic falls within this shaded region, we reject the null hypothesis.

The null hypothesis, denoted as Η0, is a statement we aim to test. It assumes no effect or difference, and it’s what we seek to nullify. In the exercise, the null hypothesis states that the population mean, Η0 μ = 50. We subject this presumption to testing with our sample data. If we find enough evidence, based on our test statistic falling in the critical region, we reject the null hypothesis. Otherwise, as in our case where the test statistic (-1.18) is greater than the critical value (-1.645), we do not reject it. This means we conclude that there isn't sufficient evidence to say the mean is less than 50.