Locating a Percentile in a Normal Distribution (Level 1) Question 32 0/0.333333333 points Identify the score corresponding to the indicated percentile. Cardiac pacemaker batteries will stop working if they are not properly calibrated and charged. Suppose that the length of time a battery operates follows a normal distribution with mean 25.0 days and standard deviation 2.0 days. At what point will 1% of the batteries fail?
21.85 days
19 days
22.31 days
20.34 days
24.98 days
Locating a Percentile in a Normal Distribution (Level 2) 0.3340 333333333 15
The correct answer and explanation is :
To find the point at which 1% of cardiac pacemaker batteries fail, we are essentially looking for the 1st percentile of a normal distribution with a:
- Mean (μ) = 25.0 days
- Standard deviation (σ) = 2.0 days
Step-by-Step Solution:
To locate a percentile in a normal distribution, we use the z-score formula:
$$
X = \mu + z\sigma
$$
Where:
- $X$ is the value (battery life) at the desired percentile
- $\mu$ is the mean
- $\sigma$ is the standard deviation
- $z$ is the z-score corresponding to the desired percentile
1. Find the z-score for the 1st percentile:
The z-score corresponding to the 1st percentile (P = 0.01) is approximately:
$$
z = -2.33
$$
(This can be found using a z-table or calculator for the standard normal distribution.)
2. Plug into the formula:
$$
X = 25.0 + (-2.33)(2.0) = 25.0 – 4.66 = 20.34
$$
✅ Correct Answer: 20.34 days
Explanation (approx. 300 words):
In problems involving normally distributed data, finding a percentile means determining the value below which a certain percentage of observations fall. In this case, we want to know the number of days at which only 1% of cardiac pacemaker batteries would have failed—meaning they lasted that long or less.
The data tells us that battery life is normally distributed with a mean of 25 days and a standard deviation of 2 days. These two parameters define the shape and spread of the normal curve.
To solve this, we first find the z-score that corresponds to the 1st percentile in the standard normal distribution. Using statistical tables or a calculator, we find that the z-score for the 1st percentile is approximately -2.33. This negative value makes sense, as we’re looking far into the lower tail of the distribution.
Next, we convert the z-score back into the actual time (X) using the formula:
$$
X = \mu + z\sigma
$$
Plugging in the numbers gives us:
$$
X = 25.0 + (-2.33)(2.0) = 20.34 \text{ days}
$$
This tells us that 1% of the batteries will fail at or before 20.34 days. This is a crucial threshold for manufacturers and medical professionals to know, as they might want to replace or check devices before this point to ensure safety.
Understanding percentiles in normal distributions helps in quality control, risk assessment, and designing safer products in real-world applications.