Questions: Question 5
For a confidence level of 98% with a sample size of 28, find the critical t value.
Transcript text: Question 5
For a confidence level of $98 \%$ with a sample size of 28 , find the critical $t$ value.
Solution
Solution Steps
Step 1: Determine the Degrees of Freedom
To find the critical \( t \) value, we first need to determine the degrees of freedom (df). The degrees of freedom for a \( t \)-distribution is calculated as:
\[
\text{df} = n - 1
\]
where \( n \) is the sample size. Given that the sample size is 28:
\[
\text{df} = 28 - 1 = 27
\]
Step 2: Identify the Confidence Level
The confidence level given is \( 98\% \). This means that the area in the two tails of the \( t \)-distribution is \( 1 - 0.98 = 0.02 \). Since the \( t \)-distribution is symmetric, the area in each tail is:
\[
\frac{0.02}{2} = 0.01
\]
Step 3: Find the Critical \( t \) Value
Using a \( t \)-distribution table or calculator, we look for the critical \( t \) value that corresponds to \( 27 \) degrees of freedom and an upper tail probability of \( 0.01 \).
The critical \( t \) value for \( 27 \) degrees of freedom and a \( 98\% \) confidence level is approximately \( 2.479 \).