An electronic product takes an average of 9 hours to move through an assembly line. If the standard deviation is 0.7 hours, what is the probability that an item will take between 8.7 and 9.2 hours to move through the assembly line?
8.7 is 0.42σ below the mean
9.2 is 0.28σ above the mean
So what does your Z table say?
To solve this problem, we need to use the standard deviation and the concept of the normal distribution.
Step 1: Convert the given values to Z-scores.
The formula to calculate the Z-score is:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Given value
μ = Mean (average)
σ = Standard deviation
For the lower bound:
Z1 = (8.7 - 9) / 0.7
For the upper bound:
Z2 = (9.2 - 9) / 0.7
Step 2: Find the probability using the Z-scores.
Once we have the Z-scores, we can use a Z-score table or a calculator to find the probabilities associated with those Z-scores.
P(8.7 ≤ X ≤ 9.2) = P(Z1 ≤ Z ≤ Z2)
Step 3: Calculate the probability using the Z-score table or calculator.
Look up the Z-scores from Step 1 in a Z-score table to find the corresponding probabilities. Subtract the probability associated with the lower Z-score from the probability associated with the upper Z-score.
P(8.7 ≤ X ≤ 9.2) = P(Z1 ≤ Z ≤ Z2)
P(Z1 ≤ Z ≤ Z2) = P(Z ≤ Z2) - P(Z ≤ Z1)
Step 4: Calculate the final probability.
Subtracting the cumulative probabilities from Step 3, we get the final probability.
P(8.7 ≤ X ≤ 9.2) = P(Z ≤ Z2) - P(Z ≤ Z1)
Now, you can use a Z-score table or a calculator to find the probabilities associated with the Z-scores, and calculate the final probability.
To find the probability that an item will take between 8.7 and 9.2 hours to move through the assembly line, we need to use the concept of the normal distribution.
The normal distribution is a statistical distribution that is commonly used to model real-world phenomena. It is characterized by its mean (average) and standard deviation.
In this case, we are given the mean of 9 hours and the standard deviation of 0.7 hours. A normal distribution can be standardized in a way that allows us to calculate probabilities using a standard normal distribution table or a statistical software.
The first step is to standardize the values of 8.7 and 9.2 using the formula:
z = (x - μ) / σ
Where:
- z is the standardized value (also known as the z-score).
- x is the value we want to standardize (8.7 and 9.2 in this case).
- μ is the mean of the distribution (9 hours).
- σ is the standard deviation of the distribution (0.7 hours).
For 8.7 hours:
z1 = (8.7 - 9) / 0.7
For 9.2 hours:
z2 = (9.2 - 9) / 0.7
After computing the z-scores, we refer to a standard normal distribution table or use statistical software to find the area under the curve between these standardized values. The area represents the probability.
Alternatively, we can use a calculator or software with built-in functions to calculate the probability directly. Using the standard normal distribution table, we can find the individual probabilities for each z-score and then subtract the smaller probability from the larger one to find the final probability.
I will use a statistical software to calculate the probability for you. Please hold on for a moment.