For a sample size of 75, find the following t-scores.
(A) The critical values needed to create a $90\%$ confidence interval.
(B) The t-score with a left probability of $0.087$.
(C) The t-score with a right tail probability of $0.025$.

• A. (A) $\left(-1.293,1.293\right)$ (B) $1.373$ (C) $-1.9925$
• B. (A) $\left(-1.293,1.293\right)$ (B) $-1.373$ (C) $1.9925$
• C. (A) $\left(-1.666,1.666\right)$ (B) $-1.373$ (C) $1.9925$
• D. (A) $\left(-1.666,1.666\right)$ (B) $1.373$ (C) $-1.9925$

Answer: (C)

Verified step by step guidance

1. Determine the degrees of freedom (df) for the t-distribution, which is the sample size minus 1: \(df = 75 - 1 = 74\).
2. For part (A), find the critical t-values for a 90% confidence interval. Since the confidence level is 90%, the total area in the two tails is \$1 - 0.90 = 0.10\(, so each tail has an area of 0.05. Use the t-distribution table or software to find the t-score corresponding to a right tail probability of 0.05 with 74 degrees of freedom. The critical values will be symmetric, so the interval is \)(-t_{0.05,74}, t_{0.05,74})$.
3. For part (B), find the t-score with a left-tail probability of 0.087. This means you need the t-value \(t\) such that \(P(T \\leq t) = 0.087\) for \(df=74\). Use the t-table or software to find this value directly.
4. For part (C), find the t-score with a right-tail probability of 0.025. This means you want the t-value \(t\) such that \(P(T \\geq t) = 0.025\) for \(df=74\). Use the t-table or software to find this value. Note that this t-score will be positive, and the corresponding left-tail probability is 0.975.
5. Summarize the results: (A) the critical values form a symmetric interval around zero for the 90% confidence level, (B) the t-score corresponds to the 8.7th percentile, and (C) the t-score corresponds to the 97.5th percentile of the t-distribution with 74 degrees of freedom.