Answer:
To find the value of X for which 70.54% of the area under the distribution curve lies to the right of it, we need to find the z-score that corresponds to this percentile and then use it to calculate the value of X.
Let z be the z-score that corresponds to the 70.54th percentile of the standard normal distribution. We can find this z-score using a standard normal table or a calculator:
z = 0.5484
This means that 70.54% of the area under the standard normal curve lies to the left of z = 0.5484, and the remaining 29.46% of the area lies to the right of it.
We can now use the formula for standardizing a normal random variable to calculate the corresponding value of X:
z = (X - μ) / σ
where μ is the mean and σ is the standard deviation.
Rearranging this formula to solve for X, we get:
X = μ + z * σ
Substituting the values given in the problem, we get:
X = 10 + 0.5484 * 4
X = 12.1936
Therefore, the value of X for which 70.54% of the area under the distribution curve lies to the right of it is approximately 12.1936.