To determine which batteries have a more uniform life in hours, we need to compare the measures of variability for the Duracell and Energizer battery samples. The chapter 3 concepts that are relevant to this analysis are standard deviation and range.
First, let's calculate the measures of variability for both samples:
For the Duracell battery sample:
Mean (μ) = 26.2
Standard Deviation (σ) = 4.1
Range = 24
For the Energizer battery sample:
Mean (μ) = (15 + 18 + 19 + 23 + 25 + 25 + 28 + 30 + 34 + 38) / 10 = 25.2
Standard Deviation (σ) = ?
Range = ?
To determine which batteries have a more uniform life, we'll compare the standard deviations and ranges of the two samples.
1. Standard Deviation:
The standard deviation measures the average amount by which individual data points differ from the mean. A smaller standard deviation indicates less variability, while a larger standard deviation indicates more variability.
For the Duracell battery sample, the standard deviation is 4.1.
For the Energizer battery sample, we need to calculate the standard deviation.
To calculate the standard deviation for the Energizer battery sample, we'll use the following steps:
1. Calculate the deviation of each data point by subtracting the mean (25.2) from each individual value.
2. Square each deviation to eliminate negative values.
3. Find the average of the squared deviations.
4. Take the square root of the average to get the standard deviation.
Following these steps, the calculations are as follows:
(15 - 25.2)^2 = 105.64
(18 - 25.2)^2 = 51.84
(19 - 25.2)^2 = 38.44
(23 - 25.2)^2 = 4.84
(25 - 25.2)^2 = 0.04
(25 - 25.2)^2 = 0.04
(28 - 25.2)^2 = 7.84
(30 - 25.2)^2 = 2.89
(34 - 25.2)^2 = 76.09
(38 - 25.2)^2 = 163.84
Sum of squared deviations = 451.16
Average of squared deviations = 451.16 / 10 = 45.116
Standard deviation (σ) = √45.116 ≈ 6.71
Comparing the standard deviations:
- The Duracell battery sample has a standard deviation of 4.1.
- The Energizer battery sample has a standard deviation of 6.71.
Based on the standard deviation, we see that the Duracell batteries have a smaller value, suggesting less variability in their life hours compared to the Energizer batteries.
2. Range:
The range measures the difference between the highest and lowest values in a sample. A smaller range indicates less variability, while a larger range indicates more variability.
For the Duracell battery sample, the range is 24.
For the Energizer battery sample, the range is 38 - 15 = 23.
Comparing the ranges:
- The Duracell battery sample has a range of 24.
- The Energizer battery sample has a range of 23.
Based on the range, we see that the Duracell batteries have a larger value, indicating more variability in their life hours compared to the Energizer batteries.
In conclusion, when comparing the measures of variability for the Duracell and Energizer battery samples, we find that the Duracell batteries have a smaller standard deviation and a larger range, which suggests a less uniform life in hours compared to the Energizer batteries.