To find the probability that an instrument produced by the machine will last between 7 and 12 months, given that the life of the instrument follows a normal distribution with a mean (μ) of 12 months and a standard deviation (σ) of 2 months, we can use the Z-score formula:
\[ Z = \frac{(X - \mu)}{\sigma} \]
where \(X\) is the value for which we want to find the Z-score.
Step 1: Find the Z-scores
-
For \(X = 7\) months: \[ Z_1 = \frac{(7 - 12)}{2} = \frac{-5}{2} = -2.5 \]
-
For \(X = 12\) months: \[ Z_2 = \frac{(12 - 12)}{2} = \frac{0}{2} = 0 \]
Step 2: Find the probabilities
Next, we will look up these Z-scores in the standard normal distribution table or use a calculator to find the probabilities.
-
For \(Z_1 = -2.5\):
- The probability \(P(Z < -2.5)\) gives us approximately \(0.0062\).
-
For \(Z_2 = 0\):
- The probability \(P(Z < 0)\) is \(0.5000\).
Step 3: Calculate the probability between the Z-scores
To find the probability that the instrument lasts between 7 and 12 months, we find:
\[ P(7 < X < 12) = P(Z < 0) - P(Z < -2.5) \]
Substituting the probabilities we found:
\[ P(7 < X < 12) = 0.5000 - 0.0062 = 0.4938 \]
Final answer
Rounding to the nearest thousandth, the probability that an instrument produced by this machine will last between 7 and 12 months is:
\[ \boxed{0.494} \]
1 answer