Stat230-Assignment 04

Stat 230 Assignement 04 Question 1 Formally prove the following: (a) (2 points) Let X be a continuous random variable that is normally distributed with mean µ and standard deviation σ . Show that approximately 95% of the area under the normal density curve is within 2 standard deviations of the mean. (b) (3 points) Prove for two continuous random variables X and Y , that V ar [ X + Y ] = V ar [ X ] + V ar [ Y ] + 2 Cov ( X, Y ) A couple potentially useful hints : - V ar [ X ] = E [ X 2 ] − ( E [ X ]) 2 = ⇒ V ar [ X + Y ] = E [( X + Y ) 2 ] − ( E [ X + Y ]) 2 - E [ X ] E [ X + Y ] = E ( E [ X ]( X + Y )) - Cov ( X, Y ) = E [ XY ] − E [ X ] E [ Y ] Question 2 (3 points) Find a 95% confidence interval for the expected value of concentration measurements taken from a chemical process. Sample measurements are 204 190 202 207 204 202 201 195 Question 3 A chemical process has produced, on the average, 800 tons of chemical per day. The daily yields for the past week are 785, 805, 790, 793, and 802 tons. (a) (4 points) Do these data indicate that the average yield is less than 800 tons and hence that something is wrong with the process? Test at the 5% level of significance. (b) (1 point) What assumptions are required for the valid use of the procedure you used to analyze these data. Question 4 In what follows, Z 1 , Z 2 , . . . , Z n is a sequence of independent standard normal random vari- ables. Also, X 1 , X 2 , . . . , X n , W 1 , W 2 , . . . , W m and V 1 , V 2 , . . . , V k are independent sets of normal random variables with common variance σ 2 and respective means µ X , µ W and µ V . S 2 X , S 2 W and S 2 V denote the sample variances of the X ’s, W ’s and V ’s, respectively. Identify the distributions (specifying values of parameters wherever possible) of the following random variables: (a). (2 points) Z 2 1 + Z 2 2 + Z 2 3 (1) (b). (3 points) 1 2 σ 2 ( X 1 − X 2 ) 2 (2) (c). (2 points) n X i =1 Z 2 i (4)