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1 - Statistics 144 Sample Midterm Solutions Fall Quarter...

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Statistics 144 — Sample Midterm – Solutions Fall Quarter 2009 PROBLEM 1 Suppose a population of N = 6 units has the following values y 1 = 10, y 2 = 6, y 3 = 9, y 4 = 11, y 5 = 17 and y 6 = 14. Suppose, we take a sample of size n = 3 (a) Give the sampling distribution of the sample total. There are ( 6 3 ) = 20 different samples. t S 25 26 27 29 30 31 32 33 ¯ y 8.33 8.67 9 9.67 10 10.33 10.67 11 f ( S ) 0.05 0.05 0.05 0.05 0.10 0.05 0.05 0.10 t S 34 35 36 37 38 40 41 42 ¯ y 11.33 11.67 12 12.33 12.67 13.33 13.67 14 f ( S ) 0.10 0.05 0.05 0.10 0.05 0.05 0.05 0.05 Total: t y = 67 Mean: ¯ y U = 11 . 17 Variance: S 2 = 14 . 97 (b) Give an estimate for the population total and show that it is unbiased. Total: t y = 67 estimated by ˆ t y = 6¯ y = 6 3 t S E ( ˆ t y ) = 6 3 · E ( y ) = 6 3 ( 25 · 0 . 05 + 26 · 0 . 05 + ... + 41 · 0 . 05 + 42 · 0 . 05 ) = 67 (c) Calculate its standard error using the sampling distribution and compare it to the formula given in class. SE ( ˆ t y ) = 36 9 ( 25 2 · 0 . 05 + 26 2 · 0 . 05 + ... + 41 2 · 0 . 05 + 42 2 · 0 . 05 ) - 67 2 = 9 . 476 Formula in class SE ( ˆ t y ) = N · ( 1 - n N ) · S n = 9 . 476 PROBLEM 2 The table below gives the number of refereed publications of a sample of 50 faculty (N=807) at a university. Assume a SRS was taken. pub 0 1 2 3 4 5 6 7 8 9 10 #fac 28 4 3 4 4 2 1 0 2 1 1
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(a) Calculate the mean number of publications per faculty.
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