If light bulbs have lives that are normally distributed with a mean of 2500 hours and a standard deviation of 500 hours, use the 68-95-99.7 rule to approximate the percentage of light bulbs having a life between 2000 hours and 3500 hours?
A. About 13.5%
B. About 50%
C. About 68%
D. About 81.5%
answer :C
agree with second answer.
To approximate the percentage of light bulbs having a life between 2000 hours and 3500 hours, we can use the 68-95-99.7 rule.
According to this rule, approximately:
- 68% of the data falls within one standard deviation of the mean
- 95% of the data falls within two standard deviations of the mean
- 99.7% of the data falls within three standard deviations of the mean
Given that the mean is 2500 hours and the standard deviation is 500 hours, we can calculate the boundaries for one standard deviation below and above the mean as follows:
Lower bound: mean - 1 standard deviation = 2500 - 500 = 2000 hours
Upper bound: mean + 1 standard deviation = 2500 + 500 = 3000 hours
Therefore, the percentage of light bulbs having a life between 2000 hours and 3500 hours is approximately:
- 68% for the range between 2000 hours and 3000 hours
- Since the range of interest (2000 hours to 3500 hours) extends beyond one standard deviation from the mean, we cannot make a precise calculation using the 68-95-99.7 rule alone.
However, we can estimate that the percentage of light bulbs falling within this wider range is likely higher than 68% but lower than 95%. Therefore, the closest estimate from the given options is C. About 68%.
To approximate the percentage of light bulbs that have a life between 2000 hours and 3500 hours, we can use the 68-95-99.7 rule, also known as the empirical rule or the three sigma rule.
According to this rule:
- 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.
Given that the mean life of the light bulbs is 2500 hours and the standard deviation is 500 hours, we can calculate the range within one standard deviation of the mean in both directions.
Lower bound = Mean - (1 * Standard Deviation)
Upper bound = Mean + (1 * Standard Deviation)
Lower bound = 2500 - (1 * 500) = 2000 hours
Upper bound = 2500 + (1 * 500) = 3000 hours
So, approximately 68% of the light bulbs have a life between 2000 hours and 3000 hours.
To further approximate the percentage of light bulbs with a life between 2000 hours and 3500 hours, we need to calculate the range within two standard deviations of the mean.
Lower bound = Mean - (2 * Standard Deviation)
Upper bound = Mean + (2 * Standard Deviation)
Lower bound = 2500 - (2 * 500) = 1500 hours
Upper bound = 2500 + (2 * 500) = 3500 hours
So, approximately 95% of the light bulbs have a life between 1500 hours and 3500 hours.
From the given options, the closest answer is C. About 68%, which is the percentage of light bulbs that falls within one standard deviation of the mean.