Questions: Use the correction for continuity and determine the normal probability statement that corresponds to the binomial probability statement. Binomial Probability P(x ≥ 102) Which of the following is the normal probability statement that corresponds to the binomial probability statement? A. P(x ≥ 101.5) B. P(x ≤ 101.5) C. P(x ≤ 102.5) D. P(x ≥ 102.5)

Solution Steps

For a binomial distribution with parameters \( n = 200 \) and \( p = 0.5 \), we calculate the mean \( \mu \) and standard deviation \( \sigma \) as follows:

\[ \mu = n \cdot p = 200 \cdot 0.5 = 100.0 \]

\[ \sigma^2 = n \cdot p \cdot q = 200 \cdot 0.5 \cdot 0.5 = 50.0 \]

\[ \sigma = \sqrt{npq} = \sqrt{50.0} \approx 7.0711 \]

Thus, we have:

• Mean \( \mu = 100.0 \)
• Standard Deviation \( \sigma \approx 7.0711 \)

To convert the binomial probability statement \( P(x \geq 102) \) to a normal probability statement, we apply the continuity correction. This gives us:

\[ P(x \geq 102) \approx P(x \geq 101.5) \]

Next, we calculate the Z-score for \( 101.5 \):

\[ z = \frac{X - \mu}{\sigma} = \frac{101.5 - 100.0}{7.0711} \approx 0.2121 \]

Final Answer