Power +, Inc produces AA batteries used in remote-controlled toy cars. The mean

Question

Power +, Inc produces AA batteries used in remote-controlled toy cars. The mean life of these batteries follows the normal probability distribution with a mean of 37 hours and standard deviation of 5.5. As a part of its quality assurance program, Power +, Inc tests samples of 25 batteries. What portion of the samples will have a mean useful life of more than 38 hours? What portion of the samples will have a mean useful life of more than 36.5 hours? What portion of the samples will have a mean useful life of between 36.5 and 38 hours?

Explanation / Answer

P(x > 38) = P((x - mean)/(SD/sqrt(n)) > (38 - mean)/(SD/sqrt(n))

= P(Z > (38 - 37)/(5.5/sqrt(25))

= P(Z > 0.91)

= 1 - P(Z < 0.91)

= 1 - 0.8186

= 0.1814

B) P(x > 36.5) =  P((x - mean)/(SD/sqrt(n)) > (36.5 - mean)/(SD/sqrt(n))

= P(Z > (36.5 - 37)/(5.5/sqrt (25))

= P(Z > -0.45)

= 1 - P(Z < -0.45)

= 1 - 0.3264

= 0.6736

C) P(36.5 < x < 38) =  P( (36.5 - mean)/(SD/sqrt(n)) < (x - mean)/(SD/sqrt(n)) < (38 - mean)/(SD/sqrt(n))

= P((36.5 - 37)/(5.5/Sqrt (25)) < Z < (38 - 37)/(5.5/sqrt (25))

= P(-0.45 < x < 0.91)

= P(Z < 0.91) - P(Z < -0.45)

= 0.8186 - 0.3264

= 0.4922

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