A population is normally distributed with mean 43.7 and standard deviation 5.2. Find the following probability. (Round your answer to four decimal places.)
p( x > 55.0 )
To find the probability that a random variable with a normal distribution is greater than a certain value, we can use the z-score formula. The z-score tells us how many standard deviations a particular value is from the mean.
The formula for the z-score is:
Z = (x - μ) / σ
Where:
- Z is the z-score
- x is the given value
- μ is the mean of the population
- σ is the standard deviation of the population
Let's calculate the z-score for x = 55.0, using the given values:
Z = (55.0 - 43.7) / 5.2
Z ≈ 2.1923
Now, we need to find the area to the right of this z-score in the standard normal distribution table (also called the Z-table). This will give us the probability that the random variable is greater than 55.0.
Looking up the value of 2.1923 in the Z-table, we find that the area to the right is approximately 0.0158.
Therefore, the probability of x being greater than 55.0 is approximately 0.0158.