Question

Several improvements have been done to a type of bateries that, according to their producer, should increase their useful life. A sample of batteries were tested before and after and the results are presented in the table below. At .05 significance level, can we conclude that the useful life of batteries has increased? Estimate the p-value.

Before	890	898	957	1020	600	859	926
After	900	1010	763	870	958	834	911

Answer 1

d = before - after

= (-10 + (-112) + 194 + 150 + (-358) + 25 + 15)/7 = -13.714

s_d = sqrt(((-10 - (-13.714))^2 + (-112 - (-13.714))^2 + (194 - (-13.714))^2 + (150 - (-13.714))^2 + (-358 - (-13.714))^2 + (25 - (-13.714))^2 + (15 - (-13.714))^2)/6) = 182.792

H₀: = 0

H₁: < 0

The test statistic t = ( - D)/(s_d/)

                             = (-13.714 - 0)/(182.792/)

                             = -0.184

DF = 7 - 1 = 6

At 0.05 significance level the critical value is t_{0.05, 6} = -1.943

Since the test statistic value is not less than the critical value(-0.184 > -1.943), so we should not reject the null hypothesis.

So there is not sufficient evidence to conclude that the useful life f batteries has increased.

P-value = P(T < -0.184)

             = 0.4300