\( X \) is a normally distributed random variable with mean 55 and standard deviation 10 . What is the probability that \( X \) is between 35 and 75 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

To find the probability that \( X \) is between 35 and 75, we can use the 68-95-99.7 rule, which states that approximately 68% of data falls within one standard deviation from the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations. First, calculate the standard deviations from the mean: - The mean (\( \mu \)) is 55. - The standard deviation (\( \sigma \)) is 10. Now, find the range: - One standard deviation below the mean: \( 55 - 10 = 45 \) - One standard deviation above the mean: \( 55 + 10 = 65 \) This means approximately 68% of the data falls between 45 and 65. Next, calculate two standard deviations: - Two standard deviations below the mean: \( 55 - 20 = 35 \) - Two standard deviations above the mean: \( 55 + 20 = 75 \) According to the 68-95-99.7 rule, about 95% of the data falls between two standard deviations from the mean. Thus, the probability that \( X \) is between 35 and 75 is approximately 0.950, or 95%.