Question: 5. (15 points) Let X1, ..., Xn be i.i.d. samples from N(ux, of) and let Y1, . .. , Ym bei.i.d. sam- ples from N(uy,

5. (15 points) Let X1, ..., Xn be i.i.d. samples from N(ux, of) and let Y1, . .. , Ym bei.i.d. sam- ples from N(uy, ov), where both ox and oy are known, but the means are unknown. Sup- pose all samples are independent. Let X = n-(X1+ .. . + Xn) and Y = m-(Y1+ .. . + Ym). (i) Show that X ~ N(ux,o' ) and Y ~ N(ux,03 /m). (ii) Show that X - Y ~ N H X - HY, o + n m (iii) Show that P HX - HYE (X-Y) -Za/2\\ O X + (X- Y) + za/2\\ OX + Y n =1-a. n m m (iv) Suppose n = 16,m =7, x =70.1, y =75.3, of =60, ov =40. Compute a 95% confidence interval for ux - uy. Can we conclude My > ux

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