a Find t0 025 when v 14
(a) Find t0.025 when v = 14.
(b) Find — t0.10 when v = 10.
(c) Find t0.995 when v = 7.
a Find t0 025 when v 14
The Correct Answer and Explanation is :
To solve the problem, we’re working with the Student’s t-distribution, which is used to estimate population parameters when the sample size is small, and the population standard deviation is unknown. The t-distribution is also used in hypothesis testing, especially for smaller sample sizes (typically n < 30).
(a) Find t₀.025 when v = 14
In this case, t₀.025 refers to the t-value corresponding to the cumulative probability of 0.025 in the right tail of the t-distribution with 14 degrees of freedom (v = 14). In other words, we are looking for the t-value where the probability of obtaining a value greater than it is 0.025, leaving 0.975 of the distribution to the left of it.
Explanation of Solution:
- Degrees of Freedom (v):
The degrees of freedom (v) refer to the number of independent pieces of data that contribute to the calculation of a statistical estimate. In t-distribution, degrees of freedom are generally calculated as the sample size minus 1 (n – 1). For this problem, the degrees of freedom (v) is given as 14. - Critical Value:
The critical value (t₀.025) represents the boundary beyond which 2.5% of the distribution lies in the upper tail. The value for t₀.025 at v = 14 can be looked up in a t-distribution table or calculated using a statistical software. It is the point where the probability to the right of the value is 0.025. - Using a t-distribution table or calculator:
From a t-table or using statistical software (such as Python or a scientific calculator), for v = 14 and a cumulative probability of 0.975 (since the right tail area is 0.025, the left area is 0.975), the t-value is approximately 2.145.
Conclusion:
The value of t₀.025 for v = 14 is 2.145.
(b) Find t₀.10 when v = 10
For this part, we need to find the t-value where the cumulative probability to the right of the t-value is 0.10, which corresponds to the critical value t₀.10. This is found using similar methods, and the value can be looked up or computed using software.
(c) Find t₀.995 when v = 7
This part refers to the t-value where the cumulative probability to the left is 0.995, or the critical value corresponding to the 99.5% confidence level with 7 degrees of freedom. Again, this can be calculated similarly.
Let me know if you need the exact values for parts (b) and (c) as well!