91Ó°ÊÓ

When dealing with small sample sizes and an unknown population variance, like in the example provided, the t-distribution is the tool to use for analyzing data. The t-distribution is similar to the normal distribution, but with fatter tails. This means it is more forgiving of outliers and provides a more realistic picture of the data's variability.
Unlike the normal distribution, the shape of the t-distribution depends on the degrees of freedom, becoming wider with fewer degrees of freedom and more similar to the normal distribution as degrees of freedom increase. In our example, with a sample size of 12, we have 11 degrees of freedom (sample size minus one). Knowing how the distribution works helps us understand where most of our data will fall and assists in calculating probabilities, known as P-values.

A one-sample t-test is a hypothesis test used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean. In our situation, we are testing the hypothesis that the sample mean is equal to 5 against the alternative that it is less than 5.
The choice to use a t-test arises because the population variance is unknown, and the sample size is relatively small (less than 30). This makes the t-distribution a suitable model for handling the variability in the data. During the test, we calculate the test statistic, often denoted as \( t_0 \). This value gauges how far the sample mean deviates from the hypothesized population mean under the null hypothesis.

The P-value is a critical aspect of hypothesis testing as it helps you determine the significance of your results. It represents the probability of obtaining a test statistic as extreme as the sample result, assuming the null hypothesis is true.
In the given problem, different P-values are calculated for different test statistics based on the t-distribution with 11 degrees of freedom. Here's a brief look at each:

• For \( t_0 = 2.05 \), you determine \( P(T_{11} < 2.05) \), which would be quite large. However, because this P-value does not align with the \( H_1: \mu < 5 \), it indicates that the null hypothesis holds.
• For \( t_0 = -1.84 \), \( P(T_{11} < -1.84) \) indicates a moderate level of significance, often enough to consider supporting \( H_1 \).
• For \( t_0 = 0.4 \), \( P(T_{11} < 0.4) \) results in a substantial P-value, suggesting that you cannot reject the null hypothesis.

A smaller P-value suggests stronger evidence against the null hypothesis, leading you towards considering the alternative hypothesis as the more likely scenario.

Degrees of freedom are a concept essential to various statistical computations, including the t-test. They refer to the number of values that can vary independently in your calculation once you've considered the required fixed parameters, like sample mean or population mean.
In a one-sample t-test, the degrees of freedom are calculated as the sample size minus one, namely \( n-1 \). For our example, with a sample size \( n = 12 \), the degrees of freedom are \( 12 - 1 = 11 \). Understanding degrees of freedom is important because it influences how the t-distribution is used. As degrees of freedom increase, the t-distribution progressively approaches the standard normal distribution.
Essentially, degrees of freedom help to tailor your test statistic to the sample size and data variability, ensuring that the inference you draw from the test is reliable and accurate given the context of the data you have.