Answer to Question #135100 in Statistics and Probability for Vishal Namdev

Question

A company has developed a new battery. The engineer in charge claims that the new
battery will have better performance than that of the old battery. To test the claim, the
company selects a simple random sample of 90 old batteries and 65 new batteries. The old
batteries ran continuously for 190 minutes with a standard deviation of 32 minutes; the
new batteries, 200 minutes with a standard deviation of 45 minutes. Test the engineer’s
claim at 0:05 level of signi…cance.

Answer 1

let:

new batteries be X₁ and old batteries be X₂

n₁=65 n₂=90

$\bar{x1}=200$ $\bar{x2}=190$

$\sigma 1=45$ $\sigma 2= 32$

$\alpha =0.05$

Solution

State the Hypothesis;

H₀: $\bar{x} 1- \bar{x} 2 >0$

H₁: $\bar{x} 1-\bar{x} 2 <= 0$

Develop the test-statistic;

$$Z= {\bar{x1}- \bar{x2} \over \sqrt{{{\sigma1}^2 \over n1}+ {{\sigma2}^2 \over n2}}}$$

$$= {200-190 \over \sqrt{{{45}^2 \over 65}+ {{32}^2 \over 90}}} = {10 \over 6.521627} = 1.5334$$

Compare test statistic with Z$\alpha$ ;

Z_0.05 = 1.645

Since Z<Z$\alpha$ we fail to reject the null hypothesis.

Therefore we conclude that there is no sufficient evidence for us to reject the engineer's claims that the new battery has better performance than the old.