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 650 hours and 1070 hours. 68% 75% (b) approximately e batteries have lifetimes between hours and hours. 95% 99.7%

Explanation:

Step1: Calculate z - scores

For \(x_1 = 650\), the z - score \(z_1=\frac{x_1-\mu}{\sigma}=\frac{650 - 860}{70}=\frac{- 210}{70}=-3\). For \(x_2 = 1070\), the z - score \(z_2=\frac{x_2-\mu}{\sigma}=\frac{1070 - 860}{70}=\frac{210}{70}=3\).

Step2: Use the empirical rule

The empirical rule for a normal distribution states that approximately 99.7% of the data lies within \(z=-3\) and \(z = 3\). So approximately 99.7% of the batteries have lifetimes between 650 hours and 1070 hours.

Step3: For part (b)

By the empirical rule, approximately 95% of the data lies within \(z=-2\) and \(z = 2\). Calculate the lower and upper bounds. Lower bound \(x_1=\mu-2\sigma=860-2\times70 = 860 - 140=720\). Upper bound \(x_2=\mu + 2\sigma=860+2\times70=860 + 140 = 1000\).

Answer:

(a) 99.7%
(b) 720, 1000