- The \(t\)-distribution with \(2\) degrees of freedom is symmetric about zero.
- The middle \(0.99\) area leaves \(0.01\) total in the tails, meaning \(0.005\) in each tail.
- The critical value \(t_0\) is found at the \(0.995\) cumulative probability: \(t_0 \approx 9.925\).
- The bounds are \(\boxed{-9.925, 9.925}\).
Explanation
- Understanding the t-distribution and tails
The Student's \(t\)-distribution is a probability distribution used to estimate population parameters when the sample size is small or the population variance is unknown. Like the normal distribution, the \(t\)-distribution is symmetric around zero.
Our goal is to find two \(t\)-values, which we can call \(-t_0\) and \(t_0\), that enclose a central area of \(0.99\). Since the total area under the probability density curve is always \(1\), the area in the two tails combined must be \(1 - 0.99 = 0.01\). Because the distribution is symmetric, each tail contains half of this remaining area:
\[ \text{Area in each tail} = \frac{0.01}{2} = 0.005 \]
This confirms our requirement to find \(t_0\) such that the area to the right of \(t_0\) is \(0.005\), and the area to the left of \(-t_0\) is also \(0.005\).
- Calculating the critical t-values
To find the critical value \(t_0\), we look for the point where the cumulative probability (the area to the left) is \(0.99 + 0.005 = 0.995\). We are working with a distribution that has \(2\) degrees of freedom. By calculating the \(0.995\) percentile for the \(t\)-distribution with \(2\) degrees of freedom, we obtain:
\[ t_0 \approx 9.925 \]
Due to the symmetry of the distribution, the lower bound is simply the negative of this value:
\[ -t_0 \approx -9.925 \]
Thus, the two \(t\)-values that divide the area under the curve into a middle \(0.99\) area and two outside areas of \(0.005\) are \(-9.925\) and \(9.925\).
- Selecting the final answer
Comparing our calculated values of \(-9.925\) and \(9.925\) with the provided options:
- \(-9.925, 9.925\)
- \(0, 9.925\)
- \(0, 6.965\)
- \(-6.965, 6.965\)
The correct choice is \(-9.925, 9.925\). Therefore, the \(t\)-values that bound the middle \(0.99\) of the area are \(\boxed{-9.925, 9.925}\).
Examples
When performing a statistical hypothesis test, choosing the correct critical value is essential for determining whether to reject a null hypothesis. For example, if you are analyzing the difference in mean performance between two small experimental groups (with a combined 2 degrees of freedom) at a 99% confidence level, you would use these critical values to establish the threshold for statistical significance. Any result falling outside the range \([-9.925, 9.925]\) would be considered statistically significant at the 1% alpha level.