D1 An apartment building bas nine floors and each floor bas four apartments. The

Question

D1 An apartment building bas nine floors and each floor bas four apartments. The building owner wants to install new carpeting in eight apartments to see how well it wears before she decides whether to replace the carpet in the entire building. The ficure below shows the floors of apartments in the building with their apartment numbers. Only the nine apartments indicated with an asterisk (*) have children in the apartment. 31 32 22* 23* 52 53 82 83 12 21 1st Floor 2nd Floor 3rd Floor 13 24 34 14 62Children 42 61 51* 54 81 41 in the apartnent 4th Floor 5th Floor 6th Floor 43 64 63 92* 93* 71 72 91 7th Floor 8th Floor 9th Floor 74* 73 84 94 (a) For convenience, the apartment building owner wants to use a cluster sampling method, in which the floors are clusters, to select the eight apartments. Describe a process for randomly selecting eight different apartments using this method. (b) An alternative sampling method would be to select a stratified random sample of eight apartments, where the strata are apartments with children might include two randomly selected apartments with children and six randomly selected apartments with no children. In the context of this situation, give one statistical advantage of selecting such a stratified sample as opposed to a cluster sample of eight apartments using the floors as clusters.

Explanation / Answer

a. In Cluster sampling the elements in the population are first divided into separate groups called clusters. Each element of the population belongs to one and only one cluster. A simple random sample of the cluster is then taken. All elements within each sampled cluster form the sample.Cluster sampling tends to provide the best results when the elements within the cluster are not alike.

So in the above case each floor is regarded as a cluster, and the idea is to use cluster sampling to select the eight different apartments. Each floor has 4 apartments, so we have to take a simple random sample of 2 floors form the nine floors so that we have our eight apartments to do the carpet testing. This can be done by assigning a random number to each floor and using a software like R to sample 2 numbers from the nine numbers using the sample() function and then using the apartments in these follors to do the carpet testing.

b. In Stratified random sampling the elements in the population are first divided into groups called strata such that each element in the population belongs to one and only one stratum. The basis for forming the strata is at the discretion of the designer of the sample. This is the statistical advantage of selecting a stratified random sample over a cluster sample. In the cluster sample the there is chance the the eight apartments selected at random might not have any children in them, so the testing process will not be complete, but in the stratified random sampling we are guaranteed of selecting at least 2 apartments with children in them so that our testing process will be complete

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