1. A study was conducted on the life of Duracell batteries and on the life of Energizer batteries. The study found that a sample of 10 Duracell batteries had a mean life in hours of 26.2 with a standard deviation of 4.1 and a range of 24. The results for a sample of 10 Energizer batteries were as follows (in hours):
15 18 19 23 25 25 28 30 34 38
Which batteries have a more uniform life in hours? Why? (Your explanation should contain direct references to chapter 3 concepts.)
Oh, batteries and their unpredictable lives! It seems like the Energizer batteries are enjoying themselves with a range of activities in their lives, while the Duracell batteries are playing it safe and sticking to a narrower range.
To determine which batteries have a more uniform life in hours, we can look at their ranges. The range is the difference between the highest and lowest values in the data set. In this case, the range for the Duracell batteries is 24, while the range for the Energizer batteries is 38-15 = 23.
So, despite being notorious for their catchy advertising slogans, Energizer batteries exhibit a wider range of life in hours compared to Duracell batteries. In statistical terms, this means the Energizer batteries have a more variable life. Maybe they're living life to the fullest with late-night parties and spontaneous adventures, while the Duracell batteries are more like cautious homebodies, adhering to a scheduled routine.
Remember, though, life is full of surprises, and statistical variability is just a measure of how much the data values tend to spread out. So, when it comes to batteries, it's always a good idea to carry an extra one just in case any of them decides to throw a surprise party for themselves!
To determine which batteries have a more uniform life in hours, we need to compare the levels of dispersion or variability in the data. In statistics, the range and standard deviation are commonly used measures of dispersion.
For the Duracell batteries, the range is given as 24, which means the difference between the maximum and minimum battery life is 24 hours. This indicates a relatively large difference in battery life within the sample.
On the other hand, the standard deviation for the Duracell batteries is 4.1 hours. Standard deviation measures the average amount by which individual values differ from the mean. A smaller standard deviation suggests less variability and a more uniform distribution.
For the Energizer batteries, we can calculate the range by subtracting the minimum value (15) from the maximum value (38), which gives us a range of 23 hours. This range is smaller compared to the Duracell batteries, indicating less variability.
To determine the standard deviation for the Energizer batteries, we need to calculate the mean first. Adding up all the values and dividing by the sample size (10) gives us a mean of 25.2 hours.
Next, we calculate the deviation of each individual value from the mean. These deviations are as follows:
15 - 25.2 = -10.2
18 - 25.2 = -7.2
19 - 25.2 = -6.2
23 - 25.2 = -2.2
25 - 25.2 = -0.2
25 - 25.2 = -0.2
28 - 25.2 = 2.8
30 - 25.2 = 4.8
34 - 25.2 = 8.8
38 - 25.2 = 12.8
Now, we square each deviation:
(-10.2)^2 = 104.04
(-7.2)^2 = 51.84
(-6.2)^2 = 38.44
(-2.2)^2 = 4.84
(-0.2)^2 = 0.04
(-0.2)^2 = 0.04
(2.8)^2 = 7.84
(4.8)^2 = 23.04
(8.8)^2 = 77.44
(12.8)^2 = 163.84
Summing up these squared deviations gives us a total of 521.8.
To find the variance, we divide this total by the sample size minus 1 (n-1). In this case, the sample size is 10, so the variance is 521.8 / 9 = 57.98.
Finally, we take the square root of the variance to obtain the standard deviation. So, √(57.98) ≈ 7.61 hours.
Comparing the ranges and standard deviations, we find that the Energizer batteries have a smaller range and standard deviation compared to the Duracell batteries. This suggests that the Energizer batteries have a more uniform life in hours, as their values cluster more closely around the mean compared to Duracell batteries.
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.