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### 204 1.2%2c1.3 sampling

Course: MATH MATH 2040, Fall 2010
School: Utah Valley University
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Introduction Statistics The 1.1 collection, organization, summarization, and analysis of information The Two Branches of Statistics- 1) Descriptive Statistics _______________________________________________________ _______________________________________________________ 2) Inferential Statistics _______________________________________________________ _______________________________________________________ Types of...

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Introduction Statistics The 1.1 collection, organization, summarization, and analysis of information The Two Branches of Statistics- 1) Descriptive Statistics _______________________________________________________ _______________________________________________________ 2) Inferential Statistics _______________________________________________________ _______________________________________________________ Types of Data1) Qualitative _________________________________________ _______________________________________________________ _______________________________________________________ 2) Quantitative ________________________________________ _______________________________________________________ _______________________________________________________ a) Continuous _______________________________________ ____________________________________________________ b) Discrete ________________________________________ 1.2 Types of Studies and Simple Random Sampling Explanatory variables - the cause variables Response variables the effect variables Their values are determined by the explanatory variables Types of studies: 1)Observational _______________________________________ _________________________________________________________ 2) Experimental _______________________________________ _________________________________________________________ Confounding - Confounding occurs when the effects of two variables on a response variable are not distinguished from each other. Lurking variables A lurking variable is a variable which is not one of the explanatory or response variables and yet it affects the response variable (confounding) Example - A plant biologist is studying the amount of sunlight received by plants in a mountain meadow and their photosynthetic and finds that plants in full sunlight have lower rates than plants in shade. A lurking variable may be the amount of moisture in the soil such that soil with more sunlight is also drier. Association does not always mean there is causation. Example. When a large number of different sized cities are compared it is found that as the number of fire hydrants increase the number of crimes also increase. Of course an increase in the number of fire hydrants does not cause an increase in the number of crimes. In this case the population of the city is a lurking variable and as the population increases so does both the number of both hydrants and crimes. In science, researchers test hypotheses and theories. To perform these tests data need to be collected and analyzed to determine which of the hypothesis are most consistent with the data. The sampling methods researchers use to collect data will be a main topic of this chapter. Specifically we will be studying: 1)simple random sampling 2)stratified sampling, 3)Systematic sampling, and 4)cluster sampling 1.3 Simple Random Sampling Definition -Assume we are taking a sample of size n from a population of size N. If every possible sample of size n has the same chance of being selected this is called a simple random simple. Note: members of the population are called units. If the units are people they are ca11ed subjects) The problem for researchers is to find a method to take a sample of size n such that all possible samples have the same chance of selection. For example, we could sample trees in forest a by moving about the forest in an unorganized fashion and haphazardly pick trees for measurements. We could also start by looking at the forest and then try to pick a. "representative group of trees for measurements. The problem with both these sampling strategies is that they rely on our judgment as to what is haphazard or representative and could result in bias in the selection. To remove the subjective aspect of the selection we use the following strategy. Assume the population is of size N = 10000 and the sample is of size 3O. (Note, the list of all the units in the population is called a frame) 1) Assign each of the units of the population unique number 2) Use a pseudorandom number generator to select 30 integers from the 10,000 possible integers. The integers selected determine the population units in the Sample. 1.4 Systematic, Stratified, and Cluster Sampling Systematic Sampling Sometimes a good frame is not available and systematic sampling is used. For example we may want to sample opinions of people living in a certain section of a city so we go down a street and sample houses. In a 1 in k systematic sample a random number, x, is chosen between l and k. If we decide to sample every tenth house k = 10. We randomly select a number between l and 10 for the starting point. Lets assume the random value chosen for x is 6, thus we Sample the 6th, 16th, 26th, 36th, etc' houses. Advantages of systematic sampling: 1) ____________________________________________________________ 2) ____________________________________________________________ 3) ____________________________________________________________ Stratified Sampling To perform stratified sampling the population is divided into nonoverlapping subpopulations called strata and then simple random samples are then taken from each stratum. Stratified sampling yields better estimates of population parameters than simple random sampling (SRS) when the variability within subgroups is relatively small while variability between subgroups is relatively large Cluster Sampling Cluster sampling, like stratified sampling divides the population into groups. However, in Contrast to Stratified Sampling each of the clusters should be similar to the entire population. For example, if we want to sample leaves in an orchard it would not be practical to get a frame of all the leaves in the orchard. Instead, we could identify the trees in the orchard, take a simple random sample of the trees to be sampled and then extensively sample the selected trees. For equal sample sizes cluster sampling is usually not as effective as simple random sampling. However, since the units in a cluster are close together we can take larger samples with cluster sampling and thus on a per unit cost basis cluster sampling can be more effective. In Summary, Cluster sampling is more effective than SRS when: 1) The variation within each cluster is about the same as in the whole population. And 2) The cost of sampling increases significantly as the distance between sampled units increases. Comparison of how Stratified sampling is conducted compared to Cluster sampling STRATIFIED Strata 1 SRS 2 SRS 3 SRS SAMPLING 4 SRS census of strata SRS within strata CLUSTER SAMPLING SRS of clusters census within clusters 1 2 3 Census 5 6 Census 7 4 Census 8
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Binomialdistribution60.95X0123456P(X)0.000000.000000.000080.002140.030540.232130.735091.00000npcumulativeprobability0.000000.000000.000090.002230.032770.264911.00000expected5.700 value0.285 variancestandard