Question 3 of 7 (s points) I Question Attempt 1 of 5 \( \checkmark 1 \) \( \checkmark 2 \) 3 4 5 6 7 Use the normal approximation to the binomial to find the probability for \( n=53, p=0.9 \), and \( X \leq 40 \). Use The Standard Normal Distribution Table. Round \( z \) value calculations to 2 decimal places and final answer to at least 4 decimal places. The probability is \( \square \) .

Let \( X \) be a binomial random variable with parameters \( n = 53 \) and \( p = 0.9 \). We wish to estimate \[ P(X \leq 40) \] using the normal approximation. 1. Compute the mean and variance of \( X \): \[ \mu = np = 53 \times 0.9 = 47.7, \] \[ \sigma^2 = np(1-p) = 53 \times 0.9 \times 0.1 = 4.77. \] Thus, the standard deviation is \[ \sigma = \sqrt{4.77} \approx 2.19. \] 2. Apply the continuity correction. We estimate \[ P(X \leq 40) \approx P(Y \leq 40.5), \] where \( Y \) is normally distributed with mean \( 47.7 \) and standard deviation \( 2.19 \). 3. Compute the \( z \)-score for \( Y = 40.5 \): \[ z = \frac{40.5 - 47.7}{2.19} \approx \frac{-7.2}{2.19} \approx -3.29. \] (Rounding to two decimal places, we have \( z \approx -3.29 \).) 4. Use the standard normal table to find \( P(Z \leq -3.29) \). From the table, \[ P(Z \leq -3.29) \approx 0.0005. \] Thus, the probability is \[ \boxed{0.0005}. \]