The diameter of a pipe is normally distributed with a mean of 0.4 inches and a variance of 0.0004. What is the probability that the diameter of a randomly selected pipe will exceed 0.44 inches?
If the variance is 0.0004, the standard deviation is the square root of that, or 0.02.
Using the computational tool at
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html ,
the probability is 9.1%
To find the probability that the diameter of a randomly selected pipe will exceed 0.44 inches, we need to calculate the z-score and then use the standard normal distribution table.
First, we calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
In this case, x = 0.44 inches, μ = 0.4 inches, and σ = √(variance) = √(0.0004) = 0.02 inches.
z = (0.44 - 0.4) / 0.02
z = 0.04 / 0.02
z = 2
Next, we look up the probability corresponding to a z-score of 2 in the standard normal distribution table. The table gives us the cumulative probability from the left-hand side, so we need to find the complement (1 - probability) to get the probability from the right-hand side.
From the table, we find that the cumulative probability for a z-score of 2 is approximately 0.9772.
Therefore, the probability that the diameter of a randomly selected pipe will exceed 0.44 inches is 1 - 0.9772 = 0.0228, or 2.28%.
To find the probability that the diameter of a randomly selected pipe will exceed 0.44 inches, we need to calculate the area under the normal curve to the right of 0.44.
Step 1: Calculate the standard deviation (SD). The standard deviation is the square root of the variance:
SD = √(variance) = √(0.0004) = 0.02
Step 2: Standardize the value of 0.44 using the z-score formula, which is (x - mean) / SD:
z = (0.44 - 0.4) / 0.02 = 0.04 / 0.02 = 2
Step 3: Look up the z-score in the Z-table or use a calculator to find the area to the left of the z-score. In this case, since we want the probability to the right of 0.44, we need to subtract the area to the left of the z-score from 1.
Using a Z-table or a calculator, the area to the left of 2 is approximately 0.9772.
Step 4: Calculate the probability to the right of 0.44 by subtracting the area to the left of the z-score from 1:
P(x > 0.44) = 1 - 0.9772 = 0.0228
Therefore, the probability that the diameter of a randomly selected pipe will exceed 0.44 inches is approximately 0.0228 or 2.28%.