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 :D
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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, also known as the empirical rule or the three-sigma rule.
According to the rule:
- Approximately 68% of the data falls within one standard deviation from the mean.
- Approximately 95% of the data falls within two standard deviations from the mean.
- Approximately 99.7% of the data falls within three standard deviations from the mean.
Given that the mean is 2500 hours and the standard deviation is 500 hours, we can calculate the z-scores for 2000 hours and 3500 hours.
For 2000 hours:
z = (2000 - 2500) / 500 = -1
For 3500 hours:
z = (3500 - 2500) / 500 = 2
Now, we can determine the percentage of light bulbs with a life between 2000 hours and 3500 hours.
Using the z-table (available online or in textbooks), we can find the area under the normal distribution curve between -1 and 2. This area represents the percentage of data falling between these z-scores.
Looking up the z-scores in the table:
- For -1, the area is approximately 0.1587.
- For 2, the area is approximately 0.9772.
To find the percentage between -1 and 2, we subtract the area to the left of -1 from the area to the left of 2:
0.9772 - 0.1587 = 0.8185
Therefore, approximately 81.85% of light bulbs have a life between 2000 hours and 3500 hours.
However, the answer choices provided are rounded percentages. Among the given options, the closest value to 81.85% is option D "About 81.5%".