To find the critical value needed for constructing a confidence interval, we typically use the t-distribution when the sample size is small (n < 30) and we do not know the population standard deviation.
The critical value can be found using a t-table or a statistical calculator, and it corresponds to the desired confidence level and degrees of freedom (df). The degrees of freedom can be calculated as df = n - 1, where n is the sample size.
Let's calculate the critical values for each case:
A) 95% confidence level with a sample size of 7:
- Degrees of freedom (df) = 7 - 1 = 6.
- For a 95% confidence level and df = 6, the critical t-value is approximately 2.447.
B) 98% confidence level with a sample size of 12:
- Degrees of freedom (df) = 12 - 1 = 11.
- For a 98% confidence level and df = 11, the critical t-value is approximately 2.718.
C) 99% confidence level with a sample size of 27:
- Degrees of freedom (df) = 27 - 1 = 26.
- For a 99% confidence level and df = 26, the critical t-value is approximately 2.779.
D) 98% confidence level with a sample size of 14:
- Degrees of freedom (df) = 14 - 1 = 13.
- For a 98% confidence level and df = 13, the critical t-value is approximately 2.650.
Summary of Critical Values:
A) 2.447
B) 2.718
C) 2.779
D) 2.650