question\nthe amount of time a certain brand of light bulb lasts is normally distributed with a mean of 2000…
question\nthe amount of time a certain brand of light bulb lasts is normally distributed with a mean of 2000 hours and a standard deviation of 35 hours. using the empirical rule, what percentage of light bulbs last between 1965 hours and 2035 hours?
Answer
Explanation:
Step1: Recall the empirical rule
The empirical rule for a normal - distribution states that about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations of the mean, and about 99.7% lies within 3 standard deviations of the mean. The formula for the range within (k) standard deviations of the mean is (\mu\pm k\sigma), where (\mu) is the mean and (\sigma) is the standard deviation.
Step2: Calculate the number of standard - deviations
We are given (\mu = 2000) hours and (\sigma=35) hours. For the lower bound (x_1 = 1965) hours, we calculate (k_1=\frac{\mu - x_1}{\sigma}=\frac{2000 - 1965}{35}=\frac{35}{35}=1). For the upper bound (x_2 = 2035) hours, we calculate (k_2=\frac{x_2-\mu}{\sigma}=\frac{2035 - 2000}{35}=\frac{35}{35}=1). The values 1965 and 2035 are 1 standard deviation below and above the mean respectively.
Step3: Apply the empirical rule
Since the values are 1 standard deviation away from the mean, by the empirical rule, the percentage of data within 1 standard deviation of the mean is 68%.
Answer:
68%