Given a t-distribution, if P(t ≥ 2.571) = 0.025, what is the probability P(t < 2.571)?
Understand the Problem
The question is asking for the probability of a t-value being less than 2.571 given that the probability of it being greater than or equal to that value is 0.025. Since probabilities sum to 1, we can find the required probability by subtracting the known probability from 1.
Answer
The probability of the t-value being less than 2.571 is $0.975$.
Answer for screen readers
The probability of the t-value being less than 2.571 is $0.975$.
Steps to Solve
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Identify the known probability
We know that the probability of the t-value being greater than or equal to 2.571 is 0.025. This can be written as:
$$ P(T \geq 2.571) = 0.025 $$ -
Use the total probability rule
The sum of probabilities for continuous distributions like the t-distribution equals 1. Therefore, we can express the probability of the t-value being less than 2.571 with the following equation:
$$ P(T < 2.571) + P(T \geq 2.571) = 1 $$ -
Substitute the known value
We can substitute the known probability into the equation we just created:
$$ P(T < 2.571) + 0.025 = 1 $$ -
Solve for the unknown probability
To find the probability of the t-value being less than 2.571, we can rearrange the equation:
$$ P(T < 2.571) = 1 - 0.025 $$
$$ P(T < 2.571) = 0.975 $$
The probability of the t-value being less than 2.571 is $0.975$.
More Information
This result indicates that there is a 97.5% chance of obtaining a t-value that is less than 2.571. In statistical terms, this suggests that 2.571 is a high value, sitting at the upper tail of the distribution.
Tips
A common mistake is misinterpreting the probabilities. Some may confuse the probability of being greater than or equal to a value with that of being less than. To avoid this, always remember that for any probability, the total must equal 1.
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