Empirical rule the authors generac generator produces voltage amounts with a mean of 125.0 volts and a standard deviation of 0.3 volt,and the voltage have a bell-shape distribution.using the empirical rule,what is the approximate percentage of voltage amounts between 124.4 volts and 125.6 volts?
You are within two standard deviations. What is the rule?
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Empirical Rule
To use the empirical rule, we need to know how many standard deviations from the mean the given range of voltages falls.
Step 1: Calculate the difference between the given range and the mean:
Upper boundary = 125.6 volts
Lower boundary = 124.4 volts
Step 2: Determine the number of standard deviations:
Upper boundary distance = (Upper boundary - Mean) / Standard deviation
Lower boundary distance = (Lower boundary - Mean) / Standard deviation
Plugging in the values:
Upper boundary distance = (125.6 - 125.0) / 0.3
Upper boundary distance = 0.6 / 0.3
Upper boundary distance = 2 standard deviations
Lower boundary distance = (124.4 - 125.0) / 0.3
Lower boundary distance = -0.6 / 0.3
Lower boundary distance = -2 standard deviations
Step 3: Determine the percentage of values within the given range:
According to the empirical rule, approximately 68% of the data falls within 1 standard deviation of the mean, approximately 95% falls within 2 standard deviations, and approximately 99.7% falls within 3 standard deviations.
Since the given range falls within 2 standard deviations from the mean, we can approximate the percentage of voltage amounts between 124.4 volts and 125.6 volts using the 95% figure.
Therefore, the approximate percentage of voltage amounts between 124.4 volts and 125.6 volts is 95%.
To use the empirical rule, we need to know the mean (μ) and standard deviation (σ) of the distribution. In this case:
- Mean (μ) = 125.0 volts
- Standard deviation (σ) = 0.3 volt
According to the empirical rule, for a bell-shaped distribution:
- Approximately 68% of the data falls within one standard deviation of the mean (μ ± σ)
- Approximately 95% of the data falls within two standard deviations of the mean (μ ± 2σ)
- Approximately 99.7% of the data falls within three standard deviations of the mean (μ ± 3σ)
Since we want to find the percentage of voltage amounts between 124.4 volts and 125.6 volts, we can calculate the number of standard deviations these values are from the mean and estimate the percentage using the empirical rule.
1. Calculate the number of standard deviations for 124.4 volts and 125.6 volts:
- For 124.4 volts: (124.4 - 125.0) / 0.3 = -2
- For 125.6 volts: (125.6 - 125.0) / 0.3 = 2
2. Determine the percentage of data within the range:
- Since the range is within two standard deviations of the mean, we can use the 95% rule.
- The percentage of data within this range is approximately 95%.
Therefore, the approximate percentage of voltage amounts between 124.4 volts and 125.6 volts is 95%.