Question 2 Sample Size (35 marks) You are requested to design a survey of households, designed to measure the key variable (among others) of travel time per day (minutes) for each person. You are asked to collect sufficient data to permit you to estimate the average travel time per person to within 15 percent at 90% confidence. From previous studies, you estimate that the average daily travel time per person per day is 80 minutes, and the variance in this statistic is 5245. The population of households from which the sample is to be drawn is 5 million. (i) What sample size would you require to achieve this level of accuracy? (ii) Suppose that your client states that this sample size is too large for the budget. As an alternative, you propose to post stratify the sample by household size. From the census, you have the information shown in Table 1 below about the number of households by household size, and your prior information about the variance of average travel time per person per day by household size is also shown in the table. (a) If your sample size remains the same as the result from question (a), calculate the weights for each stratum and also calculate the variance and standard error of the mean after post-stratified sampling is applied. (b) Based on this result, explain why post stratified sampling can reduce sample size if same level of survey accuracy is required. Table 1 Information of number of household by household size Household Size Population Estimated Variance 1 1,200,000 602 2 1,910,000 2,400 3 900,000 3,790 600,000 6,450 5+ 390,000 9,200 4 (iii) If the population was 10,000, what would the sample size be? What is the minimum population size that you think would be justified to ignore the finite population correction factor? Show by calculation how you justify this. Question 2 Sample Size (35 marks) You are requested to design a survey of households, designed to measure the key variable (among others) of travel time per day (minutes) for each person. You are asked to collect sufficient data to permit you to estimate the average travel time per person to within 15 percent at 90% confidence. From previous studies, you estimate that the average daily travel time per person per day is 80 minutes, and the variance in this statistic is 5245. The population of households from which the sample is to be drawn is 5 million. (i) What sample size would you require to achieve this level of accuracy? (ii) Suppose that your client states that this sample size is too large for the budget. As an alternative, you propose to post stratify the sample by household size. From the census, you have the information shown in Table 1 below about the number of households by household size, and your prior information about the variance of average travel time per person per day by household size is also shown in the table. (a) If your sample size remains the same as the result from question (a), calculate the weights for each stratum and also calculate the variance and standard error of the mean after post-stratified sampling is applied. (b) Based on this result, explain why post stratified sampling can reduce sample size if same level of survey accuracy is required. Table 1 Information of number of household by household size Household Size Population Estimated Variance 1 1,200,000 602 2 1,910,000 2,400 3 900,000 3,790 600,000 6,450 5+ 390,000 9,200 4 (iii) If the population was 10,000, what would the sample size be? What is the minimum population size that you think would be justified to ignore the finite population correction factor? Show by calculation how you justify this