Suppose \( Z \) follows the standard normal distribution. Calculate the following probabilities using the ALEKS calculator. Round your responses to at least three decimal places. (a) \( P(Z>-1.83)=\square \) (b) \( P(Z \leq-1.87)=\square \) (c) \( P(1.08

Ask by Morrison Pollard. in the United States

Jan 27,2025

To calculate the probabilities for the standard normal distribution using a calculator or statistical tables, we need to find the corresponding Z-scores: (a) For \( P(Z > -1.83) \): You first find \( P(Z \leq -1.83) \) and then subtract that from 1. Approximately, \( P(Z \leq -1.83) \approx 0.0336 \), so \( P(Z > -1.83) \approx 1 - 0.0336 = 0.9664 \). Therefore, rounding to three decimal places, \( P(Z > -1.83) \approx 0.966 \). (b) For \( P(Z \leq -1.87) \): Looking it up gives \( P(Z \leq -1.87) \approx 0.0307 \). Rounding this value, we get \( P(Z \leq -1.87) \approx 0.031 \). (c) For \( P(1.08 < Z < 2.19) \): Calculate this by finding \( P(Z \leq 2.19) - P(Z \leq 1.08) \). We can approximate \( P(Z \leq 2.19) \approx 0.9857 \) and \( P(Z \leq 1.08) \approx 0.8599 \). So we have \( P(1.08 < Z < 2.19) \approx 0.9857 - 0.8599 = 0.1258 \). Rounding to three decimal places gives us \( P(1.08 < Z < 2.19) \approx 0.126 \). In summary, the rounded probabilities are: (a) \( P(Z > -1.83) \approx 0.966 \) (b) \( P(Z \leq -1.87) \approx 0.031 \) (c) \( P(1.08 < Z < 2.19) \approx 0.126 \)