STAT 1060: Lecture 6 Sampling

Transcription

1 STAT 1060: Lecture 6 Sampling Chapter 11 September 23, 2011 (Chapter 11) STAT 1060 September 23, / 14

2 : Sampling (Chapter 11) Population Final marks for students in STAT 1060, Fall 2002 Size of Population: N = 326 Computed Population Mean: Task: Estimate value of Population Mean using Random Samples (Chapter 11) STAT 1060 September 23, / 14

3 Random Sampling from a Population (Chapter 11) STAT 1060 September 23, / 14

4 Sampling and Inference Column c1 contains final marks out of 100 for fall02 Stat1060 MTB > stem c1 Stem-and-leaf of C1 N = 326 Leaf Unit = (76) (Chapter 11) STAT 1060 September 23, / 14

5 MTB > hist c1 Histogram of C1 N = 326 Each * represents 2 obs. Midpoint Count 0 5 *** 10 7 **** 20 9 ***** ****** ********** *********************** ************************************** ********************************* ************************** ***************** ** MTB > descr c1 N MEAN MEDIAN TRMEAN STDEV SEMEAN C MIN MAX Q1 Q3 C (Chapter 11) STAT 1060 September 23, / 14

6 Most often we won t have access to a whole POPULATION of numbers, but only a SAMPLE from the population. In general we will only have a single sample taken from a population. Here we take 100 samples of size 30. If each possible sample is equally likely, the sample is called a SIMPLE RANDOM SAMPLE. (Chapter 11) STAT 1060 September 23, / 14

7 Random Sampling Using Minitab Draw 100 random samples from population, each of size 30 Compute the sample mean for each of the 100 samples Show the distribution of the sample means: The Sampling distribution Compute the mean value of the 100 sample means This mean of the sample means is our estimate of the Population Mean The estimate is said to be unbiased when this mean exactly equals the expected population mean When we know the shape of the sampling distribution in advance, we can estimate the value of the population mean from one sample (Chapter 11) STAT 1060 September 23, / 14

8 MINITAB: Next several lines create 100 samples of size 30 taken from c1. They are put into rows 1:100 of columns c11-c40. MTB > sample 3000 c1 c2; SUBC> replace. MTB > set c3 DATA> 100(1:30) DATA> end MTB > unstack c2 c11-c40; SUBC> subscripts c3. MTB > info COLUMN NAME COUNT C1 326 C C C C C C (Chapter 11) STAT 1060 September 23, / 14

9 In general, when sampling a population, we will have only one sample, in this case a sample of size 30. We want to make INFERENCES about the population quantity of interest (population mean in this case). Using the computer, or theoretical methods, we can study the properties of various statistics. In the following, we study the behaviour of the mean. (Chapter 11) STAT 1060 September 23, / 14

10 MINITAB: Next two lines find the means of the samples of size 30. MTB > rsum c11-c40 c41 MTB > let c41=c41/30 C (Chapter 11) STAT 1060 September 23, / 14

11 The distribution of the means is called the sampling distribution. MTB > stem c41 Stem-and-leaf of C41 N = 100 Leaf Unit = (13) MTB > descr c41 N MEAN MEDIAN TRMEAN STDEV SEMEAN C MIN MAX Q1 Q3 C (Chapter 11) STAT 1060 September 23, / 14

12 Our hope is that we have calculated a statistic which has little bias (i.e. the mean of the sampling distribution is the population parameter), and that has little variability (it doesn t change much from sample to sample). (Chapter 11) STAT 1060 September 23, / 14

13 Data Production: careful design of data production is most important prerequisite for trustworthy inference Observation versus Experiment an observational study observes experimental units (individuals) and measures variables of interest but does not attempt to influence the responses an experiment deliberately imposes some treatment on experimental units in order to observe their response experiment designer determines which treatments are assigned to which experimental units a properly designed experiment can give good evidence for causation (Chapter 11) STAT 1060 September 23, / 14

14 Sampling: an Observational Study Selecting a subset (sample) to make inferences about the population. Less expensive, more timely than a census which measures all individuals, but may be more accurate Associations may not reflect causation but may be due to confounding or lurking variables The sampling design is the method used to choose the sample Bias occurs when a sample is not representative Volunteer samples and convenience samples are notoriously bad simple random sampling is when each sample of size n has the same probability of selection - each individual has the same probability of being selected need a list of all individuals (the sampling frame) and a random number generator (Chapter 11) STAT 1060 September 23, / 14