Questions: Find the endpoint(s) on the normal density curve with the given property. Round to three decimal places. The symmetric middle area on a N(32,3) curve is about 0.50. 26.120 and 37.880 29.977 and 34.023 28.549 and 35.451 28.155 and 35.845

Solution Steps

To find the endpoints of the symmetric middle area of 0.50 on a normal distribution \( N(32, 3) \), we first determine the z-scores corresponding to the cumulative probabilities of \( 0.25 \) and \( 0.75 \). The z-scores are calculated as follows:

\[ z_{0.25} = \frac{0.25 - 0}{1} = 0.25 \]

\[ z_{0.75} = \frac{0.75 - 0}{1} = 0.75 \]

Thus, we have:

• \( z_{0.25} = 0.25 \)
• \( z_{0.75} = 0.75 \)

Next, we convert the z-scores back to the original scale using the mean \( \mu = 32 \) and the standard deviation \( \sigma = 3 \). The endpoints are calculated as follows:

\[ X_1 = \mu + z_{0.25} \cdot \sigma = 32 + 0.25 \cdot 3 = 32.75 \]

\[ X_2 = \mu + z_{0.75} \cdot \sigma = 32 + 0.75 \cdot 3 = 34.25 \]

Thus, the endpoints are:

• \( X_1 = 32.75 \)
• \( X_2 = 34.25 \)

Final Answer