Answer to Question #152032 in Statistics and Probability for JG

Question

Assuming the lasting time of laptop batteries are normally distributed with a mean of 11 hours and a standard deviation of ) 0.7 hours.
a) If we randomly select 8 batteries, what is the probability that the average lasting time of these laptops is longer than 11.5 hours.
b) In a group of 30 laptops, approximately how many of them will last less than 10 hours?
c) Knowing that a lasting time of a laptop is 20% below the third quartile, how many hours can this laptop last?

Answer 1

Let $X=$ the lasting time of laptop batteries: $X\sim N(\mu, \sigma^2)$

Given $\mu=11h, \sigma=0.7h$

a) $\bar{X}\sim N(\mu,\sigma^2/n)$

Given $n=8$

$$P(\bar{X}>11.5)=1-P(\bar{X}\leq11.5)$$

$$=1-P(Z\leq\dfrac{11.5-11}{0.7/\sqrt{8}})\approx1-P(Z\leq2.020305)$$

$$\approx1-0.978324=0.021676$$

$P(\bar{X}>11.5)=0.021676$

b)

$$P(X<10)=P(Z<\dfrac{10-11}{0.7})$$

$$\approx P(Z<-1.428571)\approx0.076564$$

$30(0.076564)=2$

(c) The third quartile Q3 is the 75th percentile of a data set. 

$$P(Z<\dfrac{X-11}{0.7})=0.75$$

$$\dfrac{X-11}{0.7}\approx0.674490$$

$$X\approx11+0.7(0.674490)=11.472143$$

$$\dfrac{80\%}{100\%}\cdot11.472143=9.18(h)$$

9.18 hours