Answer to Question #198346 in Statistics and Probability for umidu

Question

11. The battery life of a certain battery is normally distributed with a mean of 90 days and a standard deviation of 3 days.

For each of the following questions, construct a normal distribution curve and provide the answer.

a)     About what percent of the products last between 87 and 93 days?

b)     About what percent of the products last 84 or less days?

For each of the following questions, use the standard normal table and provide the answer.

c) About what percent of the products last between 89 and 94 days?

d) About what percent of the products last 95 or more days? 

Answer 1

Let $X=$ the battery life in days: $X\sim N(\mu, \sigma^2)$

Given $\mu=93\ days, \sigma=3\ days.$

a)

$$P(87<X<93)=P(X<93)-P(X\leq87)$$

$$=P(Z<\dfrac{93-93}{3})-P(Z\leq\dfrac{87-93}{3})$$

$$=P(Z<0)-P(Z\leq-2)$$

$$\approx0.5-0.022750\approx0.477250$$

$47.7250\ \%$

b)

$$P(X\leq84)=P(Z\leq\dfrac{84-93}{3})$$

$$=P(Z\leq-3)\approx0.001350$$

$0.1350\ \%$

c)

$$P(89<X<94)=P(X<94)-P(X\leq89)$$

$$=P(Z<\dfrac{94-93}{3})-P(Z\leq\dfrac{89-93}{3})$$

$$\approx P(Z<0.333333)-P(Z\leq-1.333333)$$

$$\approx0.630559-0.091211\approx0.539348$$

$53.9348\ \%$

d)

$$P(X\geq95)=1-P(X<95)$$

$$=1-P(Z<\dfrac{95-93}{3})\approx1-P(Z<0.666667)$$

$$\approx0.252493$$

$25.2493\ \%$