Xis a normally distributed random variable with mean 65 and standard deviation 20. What is the probability that X is between 2 and 78? Write your answer as a decimal rounded to the nearest thousand ii.

To find the probability that X is between 2 and 78, we need to calculate the area under the normal distribution curve between these two values.<br /><br />First, we need to standardize the values by converting them to z-scores using the formula:<br /><br />z = (X - μ) / σ<br /><br />where X is the value, μ is the mean, and σ is the standard deviation.<br /><br />For X = 2:<br />z = (2 - 65) / 20 = -63 / 20 = -3.15<br /><br />For X = 78:<br />z = (78 - 65) / 20 = 13 / 20 = 0.65<br /><br />Next, we look up the corresponding probabilities for these z-scores in the standard normal distribution table or use a calculator.<br /><br />The probability for z = -3.15 is approximately 0.0009.<br />The probability for z = 0.65 is approximately 0.7422.<br /><br />To find the probability that X is between 2 and 78, we subtract the smaller probability from the larger probability:<br /><br />P(2 < X < 78) = P(z = 0.65) - P(z = -3.15) = 0.7422 - 0.0009 = 0.7413<br /><br />Therefore, the probability that X is between 2 and 78 is approximately 0.7413.