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4. A given population of size NV =6 comprises the observations Xi = 24, 16, 32, 16, 20, 36. (i) Compute the sample means of all possible simple random samples of size n = 4 which can be selected from the population. (5 marks) (ii) Using the sample means calculated in (i), show that the sample mean x is an unbiased estimator of the population mean X. (3 marks) 5. A survey was conducted to determine the number of mobile phone SIM cards used by households among Kenyan urban residents. A simple random sample of 25 households was drawn from a residential estate comprising 800 households. The number of phone SIM cards owned in a household in the sample was as follows: N: STD 8, 6, 5, 7, 9, 4, 8, 5, 4, 6, 9, 3, 8, 9, 6, 5, 7, 9, 6, 4, 8, 4, 6, 5, 9. (i) Estimate the total number of phone SIM cards owned by the households in the residential estate. Ex (2 marks) (ii) Determine the 95% confidence limits for the population mean of the number of phone SIM cards owned by residents in the residential estate. (5 marks) (iii Determine the 95% confidence limits for the population total of the number of phone SIM cards owned by residents in the residential estate. (5 marks) 6. A survey was conducted to determine the number of cattle kept in a homestead among a pastoralist community in Kenyan. A stratified random sample of size 210 homesteads was drawn from a target population of size 3500 homesteads. This target population is divided into five strata as shown below: Stratum A B C D E Stratum size 400 600 800 1000 700 Stratum sample mean 66 75 85 72 60 Stratum standard deviation 5 6 6 (a) Determine the sample size for each stratum if the sample is to be drawn using stratified random sampling with: (i) proportional allocation; (ii) optimum allocation with fixed cost. (10 marks