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Consider a [tex]t[/tex] distribution with 23 degrees of freedom. Compute [tex]P(t \geq 1.54)[/tex]. Round your answer to at least three decimal places.
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.
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