Find the following probabilities for a sample size of 75:
(A) _____________
(B) $P(T>1.65)=$ _____________
(C) OR $T>1.65){=}_{}$ ____________

• A. (A) $0.0516$ (B) $0.0516$ (C) $0.1032$
• B. (A) $0.9484$ (B) $0.9484$ (C) $0.0516$
• C. (A) $0.0516$ (B) $0.0516$ (C) $0.0516$
• D. (A) $0.9484$ (B) $0.9484$ (C) $0.8968$

Answer: (A)

Verified step by step guidance

1. Recognize that the variable T follows a t-distribution with degrees of freedom equal to the sample size minus one, so \( df = 75 - 1 = 74 \).
2. For part (A), to find \( P(T < -1.65) \), use the t-distribution table or a calculator to find the cumulative probability to the left of \( -1.65 \) with 74 degrees of freedom.
3. For part (B), to find \( P(T > 1.65) \), note that this is the complement of the cumulative probability up to 1.65, so calculate \( 1 - P(T \leq 1.65) \) using the t-distribution with 74 degrees of freedom.
4. For part (C), to find \( P(T < -1.65 \text{ or } T > 1.65) \), recognize this is the sum of the two tail probabilities from parts (A) and (B), so add \( P(T < -1.65) + P(T > 1.65) \).
5. Use symmetry properties of the t-distribution if needed, since the t-distribution is symmetric about zero, which means \( P(T < -1.65) = P(T > 1.65) \), simplifying calculations.