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 $$ \square $$ .

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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

$$
\boxed{0.0005}.
$$
The probability is approximately 0.0005.

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