A large population (consisting of measurements of diameters of ball bearings) has mean u and standard deviation o, neither of which is perfectly known. A random sample of size n = 25 observations will...

Extracted text: A large population (consisting of measurements of diameters of ball bearings) has mean u and standard deviation o, neither of which is perfectly known. A random sample of size n = 25 observations will be taken, namely the diameters U1, U2, ·.., U25 will be observed. In other words, U1, U2, · .., U25 are independent and identically distributed random variables with u = E(U1) and Var(U1) = o². No further knowledge about the shape of the distribution is known. Define, 25 1 X 25 Ui 25 i=D1 (i) If someone says E(X 25) = 13, what information does he give you about u, 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 u? In other words, find an approximation of P(X 25 < u).="" [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="" u+="" 1,="" when="" o="">