To solve for [tex]\( P(t \geq 1.54) \)[/tex] in a [tex]\( t \)[/tex]-distribution with 23 degrees of freedom, follow these steps:
1. Understand the Distribution:
- We are dealing with a [tex]\( t \)[/tex]-distribution which is defined by its degrees of freedom. Here, the degrees of freedom (df) are 23.
2. Find the Cumulative Density Function (CDF) Value:
- The CDF gives the probability that the [tex]\( t \)[/tex]-statistic is less than or equal to a specific value. We need to find the CDF value for [tex]\( t \leq 1.54 \)[/tex]. This value is equivalent to searching for the area under the [tex]\( t \)[/tex]-distribution curve to the left of [tex]\( t = 1.54 \)[/tex].
3. Complement Rule to Find Desired Probability:
- Since we want [tex]\( P(t \geq 1.54) \)[/tex], we can use the complement rule. This is because the total area under the probability curve sums to 1.
- Thus, [tex]\( P(t \geq 1.54) = 1 - P(t \leq 1.54) \)[/tex].
4. Obtain the CDF Value:
- The CDF value for [tex]\( t \leq 1.54 \)[/tex] with 23 degrees of freedom is 0.931.
5. Complement to Find Desired Probability:
- [tex]\( P(t \geq 1.54) = 1 - 0.931 \)[/tex].
6. Calculate the Final Probability:
- [tex]\( P(t \geq 1.54) = 0.069 \)[/tex].
Thus, [tex]\( P(t \geq 1.54) = 0.069 \)[/tex].
