The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 290 hours
and a standard deviation of 6 hours. What percentage of the bulbs have lifetimes that lie within 1
standard deviation to either side of the mean?
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Approximately 34% of scores lie between the mean and one standard deviation in any normal distribution. Therefore, μ ± 1 SD contains approximately 68% of the cases.
I hope this helps. Thanks for asking.
To find the percentage of the bulbs that have lifetimes within 1 standard deviation from the mean, we can use the properties of the normal distribution.
1. Start by calculating the value of 1 standard deviation. Since the standard deviation is given as 6 hours, 1 standard deviation would be 6 hours.
2. Now, we need to find the values that lie 1 standard deviation to either side of the mean. To find the lower value, we subtract the standard deviation from the mean: 290 hours - 6 hours = 284 hours. To find the upper value, we add the standard deviation to the mean: 290 hours + 6 hours = 296 hours.
3. We now have the range of values that fall within 1 standard deviation from the mean, which is 284 hours to 296 hours.
4. To calculate the percentage of bulbs that fall within this range, we need to find the area under the normal distribution curve between these two values. Since the distribution is assumed to be normal, we can use the properties of the standard normal distribution (with a mean of 0 and a standard deviation of 1) to find this area.
5. We can use a standard normal distribution table or a calculator to find the percentage. The area under the curve between the lower value (284) and the upper value (296) corresponds to the proportion of bulbs with lifetimes within 1 standard deviation.
Using the standard normal distribution table or a calculator, you will find that the percentage of bulbs with lifetimes within 1 standard deviation to either side of the mean is approximately 68%.