The batteries from a certain manufacturer have a mean lifetime of 860 hours, with a standard deviation of 70 hours. Assuming that the lifetimes are normally distributed, complete the following statements.(a) Approximately of the batteries have lifetimes between 720 hours and 1000 hours.(b) Approximately 99.7% of the batteries have lifetimes between hours and hours.
Question
The batteries from a certain manufacturer have a mean lifetime of 860 hours, with a standard deviation of 70 hours. Assuming that the lifetimes are normally distributed, complete the following statements.(a) Approximately of the batteries have lifetimes between 720 hours and 1000 hours.(b) Approximately 99.7% of the batteries have lifetimes between hours and hours.
Solution
(a) To find the percentage of batteries that have lifetimes between 720 hours and 1000 hours, we first need to convert these lifetimes into z-scores. The z-score is calculated as (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
For 720 hours: Z1 = (720 - 860) / 70 = -2 For 1000 hours: Z2 = (1000 - 860) / 70 = 2
Looking up these z-scores in a standard normal distribution table, we find that approximately 95.4% of the batteries have lifetimes between 720 hours and 1000 hours.
(b) The statement "Approximately 99.7% of the batteries have lifetimes between hours and hours" refers to the empirical rule or the 68-95-99.7 rule. This rule states that for a normal distribution, 99.7% of the data falls within three standard deviations from the mean.
So, to find the lifetimes, we calculate as follows:
Lower limit: μ - 3σ = 860 - 370 = 650 hours Upper limit: μ + 3σ = 860 + 370 = 1070 hours
So, approximately 99.7% of the batteries have lifetimes between 650 hours and 1070 hours.
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