Question

Testing of two brands of electric toothbrushes show that Crest toothbrushes are faulty in 2 of every 25 tested. The Philips toothbrushes are faulty in 3 of every 45 test. Is there enough evidence to say with 95% confidence that the Crest toothbrushes fail less often than the Philips

Answer 1

For Crest toothbrushes:

n1 = 25, x1 = 2

p̂1 = x1/n1 = 0.08

For Philips toothbrushes:   

n2 = 45, x2 = 3

p̂2 = x2/n2 = 0.0667

α =    0.05

Null and Alternative hypothesis:  

Ho : p1 = p2  

H1 : p1 < p2  

Pooled proportion:  

p̄ = (x1+x2)/(n1+n2) = (2+3)/(25+45) = 0.0714

Test statistic:  

z = (p̂1 - p̂2)/√ [p̄*(1-p̄)*(1/n1+1/n2)] = (0.08 - 0.0667)/√[0.0714*0.9286*(1/25+1/45)] = 0.2075

p-value :  

p-value = NORM.S.DIST(0.2075, 1) = 0.5822

Decision:  

p-value > α, Do not reject the null hypothesis.

Conclusion:  

There is not enough evidence to conclude with 95% confidence that the Crest toothbrushes fail less often than the Philips.