6. A large population (consisting of measurements of diameters of ball bearings) has mean µand standard deviation o, neither of which is perfectly known. A random sample of sizen = 25 observations will be taken, namely the diameters U1, U2, ...,U2, will be observed.In other words, U1, U2, · ,U25, are independent and identically distributed random variableswith u = E(U1) and Var(U1) = o². No further knowledge about the shape of the distributionis known. Define,25T25• (i) If someone says E(X25) = 13, what information does he give you about µ, if any?(ii) If someone says o = 5, what is Var(X25)?(iii) What is the chance, approximately, that a sample of size n = 25 will have its mean,X25, smaller than the population mean µ? In other words, find an approximation ofP(X25 < H). [Hint: see if the value of o is relevant or not to answer the question.](iv) In part (iii) while approximating what theorem did you use, if any?(v) What is the chance, approximately, that a sample of size n = 25 will have its mean,X25, smaller than µ + 2, when o = 5.(vi) From a sample of size n = 25 find an approximate value of P(µ – 2< X, <µ+2),when σ- 10.

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A First Course in Probability (10th Edition)

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