Answer:
To determine the value of k such that P(k < X), we need to use the properties of the standard normal distribution.
The standard normal distribution has a mean of 0 and a standard deviation of 1. To find the probability P(k < X) for the given normally distributed random variable X with a mean of 12 and a standard deviation of 4, we need to standardize the random variable using the z-score formula:
Z = (X - μ) / σ
where Z is the standardized value, X is the value of the random variable, μ is the mean, and σ is the standard deviation.
In this case, we have:
Z = (X - 12) / 4
To find the value of k such that P(k < X), we need to find the corresponding z-score for k. This can be done by rearranging the formula:
k = Z * σ + μ
Substituting the given values into the equation:
k = Z * 4 + 12
Now, we can use a standard normal distribution table or a calculator to find the probability associated with the z-score Z and determine the corresponding value of k.
Let's say we want to find the value of k such that P(k < X) = 0.95. We can find the z-score associated with a cumulative probability of 0.95, which corresponds to an area under the standard normal curve to the left of the z-score. By looking up the z-score in the standard normal distribution table or using a calculator, we find that the z-score is approximately 1.645.
Substituting this value into the equation:
k = 1.645 * 4 + 12
k ≈ 18.58
Therefore, the value of k such that P(k < X) = 0.95 is approximately 18.58.