Sampling Design and Analysis MTH 494 Ossam Chohan Assistant Professor CIIT Abbottabad

Review of previous stuff

How to draw a SR Sample This is not as difficult as it looks  But selection is important because it leads to Investigator bias Poor estimation The procedure for selecting a Simple Random Sample is as follows: List all the units in the population (construct a sampling frame if one does not exist already), say from 1,..,N

How to select the sample In other words, give each element a unique Identification (ID) starting from 1 to the number of elements in the population N N is the total number of units in the population Using random numbers or any other random mechanism (eg Lottery or goldfish bowl), select the sample of n units from the list of N units one at a time without replacement

How to select the sample There many different random number tables. One example is given on the next slide.

Random Numbers 8442 5653 8775 1891 7666 6483 9711 6941 8092 3875 4200 6543 9063 1003 8754 2564 8890 4195 8888 6490 3476

How to use a random number table? Decide on the minimum number of digits Start anywhere in the table and going in any direction choose a number/(s) The sequence of reading the numbers should be maintained until the desired sample size is attained If a particular number is not included in your range of population values, choose another number Keep selecting the numbers till you have the required number of elements in your sample

How to use a random number table? This process of selecting a large sample using random number tables is tedious Usually we use computer generated random numbers

Advantages/Disadvantages of Simple Random Sampling Sample is easy to select in cases where the population is small Disadvantages: Costs of enumeration may be high because by the luck of the draw, the sampled units may be widely spread across the population By bad luck, the sample may not be representative because it may not be evenly spread across all sections of the population

What to do next with samples Having selected the sample, we now need to produce estimates from the sample to make certain statements about the population Usually we want to provide estimates of certain parameters in the population eg mean, medians, totals or proportions

Estimation of a Population Mean and Total We stated previously that objective of Survey sampling is to draw inferences about a population on the basis of sample evidence. There are two approaches to draw inferences: Estimation Hypothesis Testing

Suppose that a simple random sample of n accounts is drawn and we are to estimate the mean value per account for the total population of hospital records. Intuitively, we would employ the sample average To estimate µ (Parameter)

Single value of is not sufficient to estimate parameter. Goodness of estimator must be evaluated. How? is an unbiased estimator. has a variance that decreases as the sample size n increases.

Some other Estimators Estimator of the population mean µ Estimated variance of

Bound of the Error of Estimation

Estimating Population Totals Many sample surveys are designed to obtain population total. Recall that the mean for a population of size N is sum of all observations in the population divided by N. the population total---that is, the sum of all observations in the population----is denoted by the symbol τ (sound like taw). Hence Nµ = τ Intuitively, we expect the estimator of τ to be N times the estimator of µ, which it is.

Estimators for totals Estimator of the population total τ Estimated Variance of τ

Bound on the error of estimation

Estimating Population Totals Many sample surveys are designed to obtain population total. Recall that the mean for a population of size N is sum of all observations in the population divided by N. the population total---that is, the sum of all observations in the population----is denoted by the symbol τ (sound like taw). Hence Nµ = τ Intuitively, we expect the estimator of τ to be N times the estimator of µ, which it is.

Example An industrial firm is concerned about the time per week spent by scientists on certain trivial tasks. The time log sheets of a simple random sample of size n=50 employees show the average amount of time spent on these tasks is 10.31 hours with a sample variance s2= 2.25. The company employs N=750 scientists. Estimate the total number of man-hours lost per week on trivial tasks and place a bound on the error of estimation

Example Solution

Conclusion Statement Thus the estimate of total time lost is = 7732.5 hours. We are reasonably confident that the error of estimation is less than 307.4 hours

Selecting the sample size for estimating population means and totals How much observations should be included in the certain design (SRS in this case). No of observations cost money, time, talent and etc. If observations are too small then validity of results in on stake. Number of observation needed to estimate µ with a bound on the error of estimation of magnitude B is found by setting two standard deviation on the estimator, , equal to B and solving this expression for n. That is

Sample size required to estimate µ with a bound on error of estimation B

Example The average amount of money µ for hospital’s accounts receivable must be estimated. Although no prior data is available to estimate the population variance δ2, that most accounts lie within a $100 range is known. There are N=1000 open accounts. Find the sample size needed to estimate µ with a bound on the error of estimation B=$3

Example Solution

Solution Cont..

Example An investigator is interested in estimating the total weight gain in 0 to 4 weeks for N=1000 chicks fed on a new ration. Obviously, to weigh each bird would be time-consuming and tedious. Therefore, determine the number of chicks to be sampled in this study in order to estimate τ with a bound on the error of estimation equal to 1000 grams. Many similar studies on chick nutrition have been run in the past. Using data from these students, the investigator found that δ2, the population variance, was approximately equal to 36.00 (grams)2. determine the required sample size.

Conclusion The investigator, therefore, needs to weight n=126 chicks to estimate τ. The total weight gain for N=1000 chickens in 0 to 4 weeks, with a bound on the error of estimation equal to 1000 grams.

Estimation of a Population Proportion First have a look what does proportion mean?

Estimation of a Population Proportion Researchers frequently interested in the portion of population possessing a specified characteristics. E.g proportion of female voters in 2013 election. Such situations exhibit a characteristics of the binomial experiment. ????? Population proportion is represented by p and estimator as

The properties of for SRS parallel those of the sample mean y-bar if the response measurements are defined as follows: Let yi =0 if the ith element sampled does not posses the specified characteristic and yi=1 if it does. Total number of elements in the sample is Σyi. If we draw a SRS of size n, the sample proportion is the fraction of the elements in the sample that posses the characteristic of interest.

Estimators Estimator of the population proportion p: Estimated variance of

Estimators Bound on the error of estimation