Question

Let X be the lifetime of an electronic device. It is known that the average lifetime of the device is 767 days and the standard deviation is 121 days. Let x¯ be the sample mean of the lifetimes of 157 devices. The distribution of X is unknown, however, the distribution of x¯ should be approximately normal according to the Central Limit Theorem. Calculate the following probabilities using the normal approximation. (a) P(x¯≤754)= Answer (b) P(x¯≥784)= Answer (c) P(749≤x¯≤784)= Answer

Answer #1

for normal distribution z score =(X-μ)/σx | |

here mean= μ= | 767 |

std deviation =σ= | 121.0000 |

sample size =n= | 157 |

std error=σ_{x̅}=σ/√n= |
9.6569 |

a)

probability = | P(X<754) | = | P(Z<-1.35)= | 0.0885 |

b)

probability = | P(X>784) | = | P(Z>1.76)= | 1-P(Z<1.76)= | 1-0.9608= | 0.0392 |

c)

probability = | P(749<X<784) | = | P(-1.86<Z<1.76)= | 0.9608-0.0314= | 0.9294 |

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