Question
If you were constructing a 99% confidence interval of the population mean based on a sample of n = 25 where the standard deviation of the sample S = 0.05, what will the critical value be?
Answers
GPT-4o mini
To determine the critical value for a 99% confidence interval when the sample size \( n = 25 \) and the sample standard deviation \( S = 0.05 \), we can use the t-distribution since the sample size is small and the population standard deviation is not known.
1. First, we need to identify the degrees of freedom. The degrees of freedom \( df \) is calculated as:
\[
df = n - 1 = 25 - 1 = 24
\]
2. Next, we need to find the critical t-value that corresponds to a 99% confidence level. Since this is a two-tailed test, we need to find the t-value that leaves 0.5% in each tail (total of 1% for the two tails).
3. Using a t-table or a calculator, we look up the critical t-value for 24 degrees of freedom at the 0.005 level (which corresponds to 99% confidence).
The critical t-value for \( df = 24 \) at the 99% confidence level is approximately:
\[
t_{0.005, 24} \approx 2.797
\]
Therefore, the critical value for constructing the 99% confidence interval is approximately **2.797**.
1. First, we need to identify the degrees of freedom. The degrees of freedom \( df \) is calculated as:
\[
df = n - 1 = 25 - 1 = 24
\]
2. Next, we need to find the critical t-value that corresponds to a 99% confidence level. Since this is a two-tailed test, we need to find the t-value that leaves 0.5% in each tail (total of 1% for the two tails).
3. Using a t-table or a calculator, we look up the critical t-value for 24 degrees of freedom at the 0.005 level (which corresponds to 99% confidence).
The critical t-value for \( df = 24 \) at the 99% confidence level is approximately:
\[
t_{0.005, 24} \approx 2.797
\]
Therefore, the critical value for constructing the 99% confidence interval is approximately **2.797**.