Background: A machine cuts wood boards to a thickness of 25 millimeters with an accepter margin of error of +/-0.6 ml. (Assume process can be distributed by normal distribution.) The acceptable range of thicknesses for the boards is 24.4 ml to 25.6 ml, inclusive.
Because of vibration, the machine shifts from 25 ml. To check the machine, you select at random three samples of four boards and find the mean thickness of each.
You select a board. Assume the machine shifts and is cutting boards with a mean thickness of 25.4 ml and a standard deviation of .2 ml.
Question: What is the probability that you select a board that is not outside the acceptable range?
Question: You randomly select 12 boards. What is the probabiliy you select at least one boards that is not outside the acceptable range?
To answer the first question, we need to calculate the probability that a board is not outside the acceptable range of 24.4 ml to 25.6 ml.
Since the process can be assumed to follow a normal distribution, we can use the z-score formula to convert the given values into standard units.
The formula for calculating z-score is:
z = (x - μ) / σ
Where:
- x is the given value
- μ is the mean value
- σ is the standard deviation
In this case, x = 25, μ = 25.4, and σ = 0.2.
Calculating the z-score for the lower limit:
z_lower = (24.4 - 25.4) / 0.2 = -5
Calculating the z-score for the upper limit:
z_upper = (25.6 - 25.4) / 0.2 = 1
Now, we need to find the probability that a board falls within this range, which can be calculated using the standard normal distribution table or a calculator.
P(24.4 ≤ x ≤ 25.6) = P(-5 ≤ z ≤ 1)
By referring to the standard normal distribution table, the probability associated with z = -5 is approximately 0, and the probability associated with z = 1 is approximately 0.8413.
Therefore, the probability of selecting a board that is not outside the acceptable range is P(-5 ≤ z ≤ 1) = 0.8413 - 0 = 0.8413.
Moving on to the second question:
To calculate the probability of selecting at least one board that is not outside the acceptable range out of 12 boards, we can use the concept of complementary probability.
The complementary probability can be calculated by finding the probability that all 12 boards fall outside the acceptable range and subtracting it from 1.
The probability of a single board being outside the acceptable range is 1 - 0.8413 = 0.1587.
Assuming each selection is independent, we can multiply the probabilities together to find the probability that all 12 boards fall outside the acceptable range:
P(all 12 boards fall outside) = (0.1587) ^ 12
Then, we can calculate the complementary probability:
P(at least one board is not outside) = 1 - P(all 12 boards fall outside)
Therefore, P(at least one board is not outside) = 1 - (0.1587) ^ 12.
You can use a calculator to find the numerical value of this probability.