A car is designed to last an average of 12 years with a standard deviation of 0.8 years. What is the probability that a car will last less than 10 years?
To find the probability that a car will last less than 10 years, we can use the concept of z-scores and the standard normal distribution.
Step 1: Convert the given time period (10 years) into a z-score.
The formula to calculate the z-score is:
z = (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
In this case:
x = 10 years
μ = 12 years
σ = 0.8 years
Plugging these values into the formula, we get:
z = (10 - 12) / 0.8
z = -2.5
Step 2: Find the probability corresponding to the z-score.
We need to find the area under the standard normal curve to the left of the z-score -2.5. We can look up this probability in a standard normal distribution table or use a calculator to find the cumulative probability.
Using a calculator or a standard normal distribution table, we find that the cumulative probability corresponding to a z-score of -2.5 is approximately 0.0062.
Step 3: Interpret the result.
The probability that a car will last less than 10 years is approximately 0.0062, or 0.62%. This means that there is a very low chance (less than 1%) that a car will last less than 10 years based on the given information.
To solve this problem, we need to find the z-score and then use the z-table to find the corresponding probability.
Step 1: Calculate the z-score using the formula:
z = (x - μ) / σ
Where:
x = desired value (10 years)
μ = mean (12 years)
σ = standard deviation (0.8 years)
z = (10 - 12) / 0.8
z = -2.5
Step 2: Use the z-table to find the probability associated with the z-score. Looking up -2.5 in the z-table, we find that the probability is approximately 0.0062.
Step 3: Convert the probability to a percentage by multiplying by 100.
Probability = 0.0062 * 100
Probability = 0.62%
Therefore, the probability that a car will last less than 10 years is approximately 0.62%.
12-10 = 2
2 years = 2.25 std below the mean.
So, check your Z table for P(Z<-2.25)