4. Incorrect. The answer is false, not true.

To answer this question, it is necessary to calculate the probability distribution function of T.

P[T=0.1] = P[30 < X < 35] = P[(30-33)/3 < (X - μ)/σ < (35-33)/3]

P[T=0.1] = ϕ(0.67) - ϕ(-1) = 0.7486 - 0.1587 = 0.5899.

ϕ( ) is the standard normal distribution. To calculate the probabilities, it is necessary to check the tables for this distribution.

P[T=0.05] = P[(35 ≤ X < 40) U (25 < X ≤ 30)]

P[T=0.05] = P[(35-33)/3 < (X - μ)/σ < (40-33)/3] + P[(25-33)/3 < (X - μ)/σ < (30-33)/3]

P[T=0.05] = ϕ(2.33) - ϕ(0.67) + ϕ(-1) - ϕ(-2.67) = 0.3964

P[T=-0.10] = 1 - (0.5899 + 0.3964) = 0.0137

The probability distribution function of T is expressed in the following table:

 t 0.1 0.05 -0.1 P[T=t] 0.5899 0.3964 0.0137

Therefore, E[T] = (0.10)*(0.5899) + (0.05)*(0.3964) – (0.10)*(0.0137) = 0.0774.
The text passage contains a detailed calculation of a probability distribution function and the expected value of a random variable T, with the answers provided for each step.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
4. Incorrect. The answer is false, not true.

To answer this question, it is necessary to calculate the probability distribution function of T.

P[T=0.1] = P[30 < X < 35] = P[(30-33)/3 < (X - μ)/σ < (35-33)/3]

P[T=0.1] = ϕ(0.67) - ϕ(-1) = 0.7486 - 0.1587 = 0.5899.

ϕ( ) is the standard normal distribution. To calculate the probabilities, it is necessary to check the tables for this distribution.

P[T=0.05] = P[(35 ≤ X < 40) U (25 < X ≤ 30)]

P[T=0.05] = P[(35-33)/3 < (X - μ)/σ < (40-33)/3] + P[(25-33)/3 < (X - μ)/σ < (30-33)/3]

P[T=0.05] = ϕ(2.33) - ϕ(0.67) + ϕ(-1) - ϕ(-2.67) = 0.3964

P[T=-0.10] = 1 - (0.5899 + 0.3964) = 0.0137

The probability distribution function of T is expressed in the following table:

 t 0.1 0.05 -0.1 P[T=t] 0.5899 0.3964 0.0137

Therefore, E[T] = (0.10)*(0.5899) + (0.05)*(0.3964) – (0.10)*(0.0137) = 0.0774.
user
The text passage contains a detailed calculation of a probability distribution function and the expected value of a random variable T, with the answers provided for each step.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS