Final answer:
The probability P(x≤126) with a mean of 120 and a standard deviation of 6 can be found by standardizing the variable to a Z-score and then using the standard normal distribution table or a calculator to find the corresponding probability.
Step-by-step explanation:
The question involves finding the probability of a random variable with a normal distribution. Specifically, you're asked to find P(x≤126) when the mean (μ) is 120 and the standard deviation (σ) is 6. To solve this, you need to use the properties of the normal distribution and the standard normal distribution table or a calculator with normal distribution functions.
First, you need to standardize the variable x using the formula Z = (X - μ) / σ. Here, Z is the standard normal variable, X is 126, μ is 120, and σ is 6. After finding the Z-score, you then look up this value in the standard normal distribution table or use a calculator to find the area to the left of that Z-score. This area represents the probability P(x≤126).
To standardize: Z = (126 - 120) / 6 = 1. You then find the area to the left of Z = 1 which gives you the probability you're looking for.