Use the ALEKS calculator to solve the following problems. \[ \begin{array}{l}\text { (a) Consider a } t \text { distribution with } 10 \text { degrees of freedom. Compute } P(-1.11

Ask by Howell Bowers. in the United States

Feb 02,2025

Sure, let's solve both problems involving the \( t \)-distribution with the given degrees of freedom (df). We'll use the properties of the \( t \)-distribution to find the required probabilities and critical values. --- ### **Problem (a)** **Given:** - Degrees of freedom (\( df \)) = 10 - We need to compute \( P(-1.11 < t < 1.11) \). **Solution:** The probability \( P(-1.11 < t < 1.11) \) is the area under the \( t \)-distribution curve between \( t = -1.11 \) and \( t = 1.11 \). Since the \( t \)-distribution is symmetric around 0, this can be calculated as: \[ P(-1.11 < t < 1.11) = 2 \times P(t < 1.11) - 1 \] To find \( P(t < 1.11) \) for \( df = 10 \), we can use a \( t \)-distribution calculator or software. Using a \( t \)-distribution calculator: \[ P(t < 1.11) \approx 0.867 \] Therefore, \[ P(-1.11 < t < 1.11) = 2 \times 0.867 - 1 = 0.734 \] **Answer:** \[ P(-1.11 < t < 1.11) = 0.734 \] --- ### **Problem (b)** **Given:** - Degrees of freedom (\( df \)) = 16 - We need to find the critical value \( c \) such that \( P(t \geq c) = 0.10 \). **Solution:** To find the value of \( c \) such that the area to the right of \( c \) under the \( t \)-distribution curve is 0.10, we look for the 90th percentile (\( 1 - 0.10 = 0.90 \)) of the \( t \)-distribution with \( df = 16 \). Using a \( t \)-distribution table or calculator: \[ c \approx 1.337 \] **Answer:** \[ c = 1.337 \] --- ### **Final Answers** **(a)** \( P(-1.11 < t < 1.11) = 0.734 \) **(b)** \( c = 1.337 \)