The lifetime of a 2-volt non-rechargeable battery in constant use has a normal distribution with a mean of 516 hours and a standard deviation of 20 hours. Ninety percent of all batteries have a lifetime less than
To find the lifetime of a 2-volt non-rechargeable battery in constant use that corresponds to the 90th percentile (in other words, the time at which 90% of all batteries have a shorter lifetime), we need to use the concept of z-scores.
A z-score is a measure of how many standard deviations an observation or data point is from the mean. In this case, we want to find the z-score that corresponds to a cumulative probability of 90%.
To find the z-score, we can use the formula:
z = (X - μ) / σ
Where:
- X is the value we want to find (the battery lifetime)
- μ is the mean of the distribution (516 hours)
- σ is the standard deviation of the distribution (20 hours)
- z is the z-score
Substituting in the values we have:
z = (X - 516) / 20
To find the z-score that corresponds to a cumulative probability of 90%, we can look up this value in a standard normal distribution table. The table provides the area under the curve to the left of a given z-score. We need to find the z-score that corresponds to an area of 0.90.
Looking up the value in the standard normal distribution table, we find that a cumulative probability of 0.90 corresponds to a z-score of approximately 1.28.
Now we can solve for X:
1.28 = (X - 516) / 20
Rearranging the equation to solve for X:
X - 516 = 1.28 * 20
X - 516 = 25.6
X = 25.6 + 516
X = 541.6
Therefore, the lifetime of a 2-volt non-rechargeable battery in constant use, at which point 90% of all batteries have a shorter lifetime, is approximately 541.6 hours.
To find the lifetime value below which 90% of all batteries fall, we need to calculate the z-score corresponding to the 90th percentile using the standard normal distribution table or a calculator.
The z-score corresponding to a percentile can be found using the formula:
z = (X - μ) / σ
Where:
X is the value we want to find the percentile for,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
In this case, X is the value we want to find (lifetime in hours), μ is the mean (516 hours), and σ is the standard deviation (20 hours).
Let's calculate the z-score first:
z = (X - μ) / σ
z = (X - 516) / 20
To find the z-score for the 90th percentile, we can use the inverse of the standard normal distribution function. In this case, we want to find the z-score corresponding to a cumulative probability of 0.90.
Using a standard normal distribution table or a calculator, we find that the z-score for a cumulative probability of 0.90 is approximately 1.28.
Now, we substitute this value into the equation:
1.28 = (X - 516) / 20
Next, we can solve for X:
20 * 1.28 = X - 516
25.6 = X - 516
X = 516 + 25.6
X = 541.6
Therefore, 90% of all non-rechargeable batteries have a lifetime less than 541.6 hours.