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Q. 1
(a)Describe briefly the difference between i) Probability and Non-probability sampling. ii) Stratified and Cluster sampling [4]
(b)A local community is stratified in four blocks. If we wish to select a stratified random sample of size n = 40 by proportional allocation on the basis of number of houses in each block. Block A B C D No. of households 144 162 198 216 Calculate the sample size allocated to each block. [4]
(c)A population consists of 2, 4, 6, 8 and 10. i) Draw all possible samples of size n = 2 without replacement. ii) Calculate the mean of each sample and verify that μ x ˉ = μ and σ x ˉ = σ n N − n N − 1 \mu_{\bar{x}} = \mu \quad \text{and} \quad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\sqrt{\frac{N-n}{N-1}} μ x ˉ = μ and σ x ˉ = n σ N − 1 N − n iii) Between what two values would you expect at least 98 of the sample means to fall? [12]
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Q. 2
(a)Describe the following i) Point estimator ii) Confidence level iii) Maximum likelihood Estimation [6]
(b)If X1,X2 and X3 are a random sample from a normal population with the mean μ and the variance σ2 , what is the relative efficiency of the estimator T1=X1+2X2+X3 with respect to ˉT2=Xˉ ? [6]
(c)There is a proposal under consideration to build an overhead bridge in a locality. As a part of feasibility and acceptability of this proposal, a poll is taken among the residents of the city and its suburbs. If 2400 of 4000 city residents favour the proposal and 1500 of 2000 suburban residents favour it, find a 90% confidence interval for the true difference between the proportion of city and suburban residents who favour the proposal to construct the overhead bridge. [8]
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Q. 3
(a)Define Type-I error and Type-II error. [2]
(b)A sample of 25 observations from a normal population with σ=3 , is selected at random. Test the hypothesis H0:μ=67 against H1:μ>67 at 5% level of significance. [8]
(c)Given two random samples of size n1=9 and n2=16 , from two independent normal populations, with x ˉ 1 = 75 , x ˉ 2 = 60 , s 1 = 13.61 \bar{x}_1 = 75, \quad \bar{x}_2 = 60, \quad s_1 = 13.61 x ˉ 1 = 75 , x ˉ 2 = 60 , s 1 = 13.61 , test the hypothesis at the 10% level of significance that μ1=μ2 against the alternative that μ1>μ2 . Assume that the populations have equal variances. [10]
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Q. 4
(a)In the context of analysis of variance, define the following: i) Main effects ii) Interaction effect [4]
(b)Determinations of yields of a process with four treatments are given: Treatments 1 2 3 4 Yields 11 6 8 14 4 4 6 27 4 3 4 8 5 6 11 18 i. Test the hypothesis that no differences exist among the four treatments at α=0.05 . ii. Apply Least Significant Difference test to identify the pairwise significant differences at 5% level of significance. [16]
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Q. 5
(a)The following is percentage distribution by income level and ownership of a random sample of 400 families in the city of Lahore. Monthly income Less than Rs.60,000 Rs.60,000 to Rs.100,000 More than Rs.100,000 Home Owner 5% 25% 20% Renter 15% 25% 10% Test the hypothesis that the home ownership is independent of the family income level, using 1% level of significance. [10]
(b)Given the two samples below, test the null hypothesis that the population medians are equal against the alternative that M 1 <M 2 , at α=0.05 by applying the Wilcoxon rank-sum test. Sample 1 26, 25, 38, 33, 42, 40, 44, 26, 25, 43, 35, 48, Sample 2 44, 30, 34, 47, 35, 46, 35, 47, 48, 34, 32, 42, 37, 43, 49, 46, 47 [10]
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Q. 6
(a)Compute the consumer price index number for 2020 with 2015 as base for the following data. Use as weights (i) quantities consumed in the base year (ii) the values in the base year. Article 2015 2020 Quantity Price (Rs.) Quantity Price (Rs.) Food 50 kg 180 - 265 Cloth 30 metre 260 - 280 Electricity 75 units 25 - 30 Rent 1 room 3000 - 3750 Miscellaneous 34 units 50 - 70 [8]
(b)Given the following data. Year Quarters I II III IV 2015 112 125 129 110 2016 119 132 147 115 2017 120 142 150 118 2018 128 151 162 125 i) Fit a linear trend to the annual averages. ii) Calculate quarterly trend values from the trend equation obtained in part (i). [12]
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Q. 7
(a)Describe the functions of Pakistan Bureau of Statistics. [8]
(b)Calculate the crude death rate and the standardized death rate for the data: Age (years) Population Number of Deaths Standard Population ('000) Males Females Males Females Males Females 0-14 2,110 2,010 30 27 59 55 5-14 3,340 3,230 6 8 109 102 15-34 7,320 7,310 16 20 177 180 35-59 7,960 8,750 70 57 121 122 60 & over 3,240 4,280 196 230 34 41 [12]
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Q. 8
(a)The following data were computed from personal records of a manufacturing firm X: number of years of service Y: weekly wage rate n = 23 , ∑ X = 2433 , ∑ n=23,∑X=2433,∑X2=281019,∑Y=4245, ∑ ∑Y2=841786 and ∑XY=482788.i.FitaleastsquaresregressionlineY=Y = \alpha + \beta X + \varepsilonii.TestthehypothesisH_0 : \beta = 0$ [10]
(b)The price of rice (X) and price of wheat (Y) at 243 shops are recorded with the results: ∑ X = 5442.2 , ∑ X=5442.2,∑X2=122155.04,∑Y=4019.6,∑∑Y2=66588.92 and ∑XY=90113.83.i.TestthehypothesisH_0: ho = 0$ . ii. Calculate 95% confidence interval for the true correlation coefficient between X and Y. [10]
The 2021 PMS Punjab Statistics paper set by the PPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.
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