Question

Let x be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12 hour fast. Assume that for people under 50 years old, x has a distribution that is approximately normal, with mean μ = 60 and estimated standard deviation σ = 44. A test result x < 40 is an indication of severe excess insulin, and medication is usually prescribed. (a) What is the probability that, on a single test, x < 40? (Round your answer to four decimal places.) (b) Suppose a doctor uses the average x for two tests taken about a week apart. What can we say about the probability distribution of x? Hint: See Theorem 6.1. The probability distribution of x is approximately normal with μx = 60 and σx = 44. The probability distribution of x is not normal. The probability distribution of x is approximately normal with μx = 60 and σx = 22.00. The probability distribution of x is approximately normal with μx = 60 and σx = 31.11. What is the probability that x < 40? (Round your answer to four decimal places.) (c) Repeat part (b) for n = 3 tests taken a week apart. (Round your answer to four decimal places.) (d) Repeat part (b) for n = 5 tests taken a week apart. (Round your answer to four decimal places.) (e) Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as n increased? Yes No Explain what this might imply if you were a doctor or a nurse. The more tests a patient completes, the stronger is the evidence for excess insulin. The more tests a patient completes, the weaker is the evidence for lack of insulin. The more tests a patient completes, the weaker is the evidence for excess insulin. The more tests a patient completes, the stronger is the evidence for lack of insulin.

Answer #1

a)

probability
=P(X<40)=(Z<(40-60)/44)=P(Z<-0.45)=0.3264 |

b)

sample size =n= | 2 |

std error=σ_{x̅}=σ/√n= |
31.11 |

**The probability distribution of x is approximately
normal with μx = 60 and σx = 31.11.**

probability
=P(X<40)=(Z<(40-60)/31.113)=P(Z<-0.64)=0.2611 |

c)

sample size =n= | 3 |

std error=σ_{x̅}=σ/√n= |
25.40341 |

probability
=P(X<40)=(Z<(40-60)/25.403)=P(Z<-0.79)=0.2148 |

d)

sample size =n= | 5 |

std error=σ_{x̅}=σ/√n= |
19.68 |

probability
=P(X<40)=(Z<(40-60)/19.677)=P(Z<-1.02)=0.1539 |

e)

Yes The more tests a patient completes, the stronger is the evidence for excess insulin.

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