Question

We have a random sample of size 17 from the normal distribution N(u,02) where u and o2 are unknown. The sample mean and variance are x = 4.7 and s2 = 5.76 (a) Compute an exact 95% confidence interval for the population mean u (b) Compute an approximate (i.e. using a normal approximation) 95% confidence interval for the population mean u (c) Compare your answers from part a and b. (d) Compute an exact 95% confidence interval for the population variance o2 (e) Use the fact that 52 « No2, 204) to compute an approximate 95% confidence interval for the population variance o2 (f) Compare your answer from part (d) and (e)

Answer 1

(a) For exact confidence interval of a ă t set it is n-1, 1-2/ where th-1, 1-4/2 n is the 1-2/ th MONDO quantile of thoi distribution now x = 0.05 -) 1-2% - 0.975, and n=17, =) t = 2.12, 16, 0.975 (I is 4.7 2:12 * 15.76 ✓ 16 =) (I is (3:428, 5.972)

where Zidh is ot i f ZI 1-dly th the l-4 N/0,1) distribution of x² = (n-1)5² (b) Approximate CI is S Sn. of the quaulile the approximate CI is (3.524,5.876). CI in (b) is contained in CI is (a). P ( 1, 0.025 é (n-1)5² (C) so 2 do Now X nal Lt 4-1,0.975 now o 2 0.95 2 where Knie a is the ath quantile of x. n-1 wheede n = 17 =) n-l16 ( VI (n-1)52 P ( (m-1352 so 216,0.975 2² 2 16, 0.025 ((n-1) 5? =) 95% CI for o?is (n-1) 5 X16,0.975 276,0.025 =) 95 %. (I for olis (3-195, 13. 342)

e) Now s' approximally follow N/0², 204 2004) s²-g² N (011), 264 nal ~ N(011), s²-0² 13n-1 =) P P ( 20.025 < 20.975) < (5²_6²)55-1 652 =o.95 -) P ( 20.025 € s² Vn- 1 ) < 20.975) rz ZO.97; t =) P ( 20.025 + vo < all +1 ) o 2 18 2² is 95% CI CI of s² Zo:97)+1 +1 s² Zo.025 Vo 18.759), Jo ( 3. 4028

Now (£) the confidence interval in part (d) is shifted to left as compared to coufidence interval ine Further confider dence interval in (el is mider too as compared to (d)