The lifetimes of light bulbs of a particular type are normally distributed with a mean of 270 hours and a standard deviation of 11 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 questions.
I believe the "68-95-99.7 rule" means the percentage of observations within 1-2-3 standard deviations from the mean.
Try to apply the rule to answer the given question.
95%
To apply the 68-95-99.7 rule, we need to calculate the range within 2 standard deviations of the mean.
Step 1: Calculate the range within 1 standard deviation:
One standard deviation from the mean accounts for approximately 68% of the data.
Lower limit: Mean - 1 standard deviation = 270 - 11 = 259
Upper limit: Mean + 1 standard deviation = 270 + 11 = 281
Step 2: Calculate the range within 2 standard deviations:
Two standard deviations from the mean account for approximately 95% of the data.
Lower limit: Mean - 2 standard deviations = 270 - (2 * 11) = 248
Upper limit: Mean + 2 standard deviations = 270 + (2 * 11) = 292
Therefore, approximately 95% of the light bulbs will have lifetimes that lie within 2 standard deviations of the mean (between 248 and 292 hours).
To solve this question using the 68-95-99.7 rule, we need to identify the percentage of bulbs whose lifetimes lie 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.
Since we are interested in the bulbs whose lifetimes fall within 2 standard deviations, we can conclude that approximately 95% of the bulbs will have lifetimes within the range of (mean - 2 * standard deviation) to (mean + 2 * standard deviation).
Using the given mean of 270 hours and the standard deviation of 11 hours, we can calculate the range within which approximately 95% of the bulbs will lie:
Range = (270 - 2 * 11) to (270 + 2 * 11)
Simplifying this, we get:
Range = 248 to 292
Therefore, approximately 95% of the bulbs will have lifetimes that lie within the range of 248 hours to 292 hours.