BASIC STATISTICS — DU B.Sc (hons) Statistics Previous Year Question Papers

Prove that if X follows F- distribution with (m, n) d. f.and Y follows the F- distribution with (n, m) d. f., then for every a > 0, p (X < a) + p(y < (a) Let X,, Xr,..., X, be a random sample from N (p,o2) and / and s2 the sample mean and sum of the squares of the deviations from the mean respectively. Let X'-N (p,o2) and assume that X,, X2,..., X,, X' are independent, obtain the samp' ,fg distribution of U _(x'-x) n(n-1) (b) Define F statistic. In one sample of 8 observations, the sum of squares of deviations of the observations from the sample mean is 84.4 anti in the other sample of l0 observations it was 102.6. Test whether this difference is significant at 5% level of significance. Given that Fo.o, U ,91 = 3.29. Write short notes on any three of the following: (i) Null hypothesis and level of significance. ) (ii) Relation between chi-square and F distributions. (iii) Type I and Type II errors. (iv) Mean deviation about mean for the t-distribution with n d.f. (5,5,5) Sampling Distributions COLLE e ark s Write your Roll. No. on th of this question paper. (a) Discuss the test of significance for difference of proportions for large samples. (b) Explain the term standard error and sampling distribution. Show that in a series of n independent trials with constant probability p of success, the standard error of the proportion of success is pq n where q : 1-p (7,8) (a) A sample of 15 values shows that the s.d. is 6.4' Is this compatible with the hypothesis that the sample is from a normal population with s.d. 5 at 5% level of significance? Justify your answer. (b) Show that chi-square distribution tends to normal distribution for large n. (7,8) (a) Define Fisher's t statistic. Find mean, variance and P, of t distribution with n d'f.. (b) Show that the probability curve ofF distribution is positively skewed. If X is a F-variate with 2 and n d.f.. Show that (a) Derive the 72 test of goodness of fit of a theoretical distribution to an observed frequency distribution with assumptions. The following table gives the number of aircraft accidents that occurred during the seven days of the week. Find whether the accidents are uniformly distributed over the week. ( 2.2 0.05 for 6 d.f. =12.592) Mon. Tue. Wed. Thur. Fri. Sat. Sun ) (b) Define Chi-square variate with I d.f. Derive p.d.f. of a Chi-square distribution with n d.f.. (7,8) (a) The average hourly wage of a sample of 150 workers in a plant 'A' was Rs.2.56 with a standard deviation ofRs. 1.08. The average hourly wage of a sample of 200 workers in plant 'B' was Rs. 2.87 with standard deviation of Rs. 1.28. Can an applicant safely assume that the hourly wages paid by plant 'B' are higher than those paid by plant'A'? (b) For the t-distribution with n d. f., prove that (a) Obtain the mode of F distribution with ni and n2 degrees of freedom, also show that it lies between (b) If X has an F distribution with (nr nr) degrees of freedom, find the distribution of l/X .