the authors's generac generator produces voltage amounts with a mean of 125.0 volts and a standard deviation of 0.3 volts, and the voltages have a bell-shaped distribution. using the empiral rule, what is the percentage of voltage amounts between 124.4 and 125.6 volts?
Please help
Empirical rule is 68-95-99.7, which means approximately 68% is between 1 standard deviation about the mean, approximately 95% is between 2 standard deviations about the mean, and approximately 99.7% is 3 standard deviations about the mean.
124.4 is two standard deviations below the mean and 125.6 is two standard deviations above the mean. (Remember 1 standard deviation is equal to 0.3 volts.)
I'll let you take it from here.
To find the percentage of voltage amounts between 124.4 and 125.6 volts using the empirical rule, we can use the concept of standard deviations.
The empirical rule, also known as the 68-95-99.7 rule, states that for a bell-shaped or normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, the mean voltage is 125.0 volts with a standard deviation of 0.3 volts.
First, let's calculate the range within one standard deviation of the mean:
Lower limit = Mean - Standard deviation = 125.0 - 0.3 = 124.7 volts
Upper limit = Mean + Standard deviation = 125.0 + 0.3 = 125.3 volts
This means that approximately 68% of the voltage amounts will fall between 124.7 and 125.3 volts.
Next, let's calculate the range within two standard deviations of the mean:
Lower limit = Mean - (2 * Standard deviation) = 125.0 - (2 * 0.3) = 124.4 volts
Upper limit = Mean + (2 * Standard deviation) = 125.0 + (2 * 0.3) = 125.6 volts
This means that approximately 95% of the voltage amounts will fall between 124.4 and 125.6 volts.
Therefore, the percentage of voltage amounts between 124.4 and 125.6 volts, using the empirical rule, is approximately 95%.