Let
x
represent the full height of certain species of tree. Assume that
x
is normally distributed with a mean of 196.3 feet and a standard deviation of 9.9 feet.
Find the probability that the full height of a randomly selected tree is less than 201.3 feet.
Enter your answer as a number accurate to 4 decimal places.
To find the probability that the full height of a randomly selected tree is less than 201.3 feet, we need to calculate the area under the normal distribution curve to the left of 201.3 feet.
1. Standardize the value: To do this, we calculate the z-score using the formula: z = (x - μ) / σ, where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation. In this case, x = 201.3, μ = 196.3, and σ = 9.9.
z = (201.3 - 196.3) / 9.9
z = 0.505
2. Calculate the probability: We can now look up the probability corresponding to the z-score of 0.505 in the standard normal distribution table or use a calculator to find the cumulative probability.
Using a calculator, the probability can be found by using the cumulative distribution function (CDF) of the standard normal distribution.
P(X < 201.3) = 0.6933 (rounded to 4 decimal places)
So, the probability that the full height of a randomly selected tree is less than 201.3 feet is 0.6933.