Area Calculation for Normal Distribution

Explanation

z = (x-u)/sigma. z1 = 400 - 400/50 = 0 and z2 = 482 - 400/50 = 1.64. Therefore for z = 1.64, the area under the curve is 0.4495.

To find the area under the normal curve between 400 and 482, we need to calculate the probability associated with that range. Since the distribution approximates a normal distribution, we can use the properties of the standard normal distribution to solve this problem.

The standard normal distribution has a mean of 0 and a standard deviation of 1. In this case, we are given the mean and standard deviation of the distribution of efficiency ratings: mean = 400 and standard deviation = 50. To use the standard normal distribution, we need to standardize our values by converting them to z-scores.

To calculate the z-score for 400, we use the formula:

z = (x - μ) / σ

where z is the z-score, x is the value we want to standardize, μ is the mean of the distribution, and σ is the standard deviation.

For 400: z = (400 - 400) / 50 z = 0

Similarly, for 482: z = (482 - 400) / 50 z = 1.64

Now that we have the z-scores for the two values, we can use the standard normal distribution table or a calculator to find the probabilities associated with these z-scores.

Using the standard normal distribution table, we can find the probability associated with a z-score of 0, which is the probability of getting a value less than or equal to 400. This probability is 0.5000.

Next, we find the probability associated with a z-score of 1.64, which is the probability of getting a value less than or equal to 482. This probability is 0.9495.

To find the area between 400 and 482, we subtract the probability associated with the lower value from the probability associated with the higher value:

0.9495 - 0.5000 = 0.4495

Therefore, the area under the normal curve between 400 and 482 is approximately 0.4495.

Among the given answer choices, the closest option is D. 0.4495.