Question

Find the critical value needed to construct a confidence interval with the given sample size:

A) 95% with a sample size of 7
B) 98% with 12
C) 99% wirh 27
D) 98% with 14

Answer 1

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