The lifetimes of projector bulbs of a particular type are normally distributed with a mean of 470 hours and a standard deviation of 15 hours. What percentage of the bulbs has lifetimes that lie within 2 standard deviations of the mean on either side?
Apply the 68-95-99.7 rule to this question.
The "68-95-99.7 rule" represents the two-tail probability of data within ±1,2 and 3 standard deviations respectively.
For example, the probability of picking a value X from a normal distribution within the range μ-σ<Xμ+σ is approximately 68%.
Using this rule, you will not need the normal distribution table to provide an answer to the question.
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To answer this question using the 68-95-99.7 rule, we need to determine the percentage of bulbs that have lifetimes within 2 standard deviations of the mean on either side.
According to the 68-95-99.7 rule:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
In this case, we are interested in the percentage of bulbs within 2 standard deviations of the mean, which is approximately 95%.
To calculate the range within 2 standard deviations of the mean, we can multiply the standard deviation (15 hours) by 2, which gives us a range of 30 hours.
To find out the specific range of lifetimes within 2 standard deviations of the mean, we need to subtract and add the range (30 hours) from the mean (470 hours).
Lower limit = mean - 2 * standard deviation = 470 - 2 * 15 = 470 - 30 = 440 hours
Upper limit = mean + 2 * standard deviation = 470 + 2 * 15 = 470 + 30 = 500 hours
Therefore, the lifetimes of the bulbs that lie within 2 standard deviations of the mean range from 440 hours to 500 hours.
Thus, the percentage of the bulbs whose lifetimes lie within 2 standard deviations of the mean is approximately 95%.