Sampling (Statistics)

Transcription

1 Systems & Biomedical Engineering Department SBE 304: Bio-Statistics Random Sampling and Sampling Distributions Dr. Ayman Eldeib Fall 2018 Sampling (Statistics) Sampling is that part of statistical practice concerned with the selection of a subset of individual observations within a population of individuals intended to yield some knowledge about the population of concern, especially for the purposes of making predictions based on statistical inference. Sampling is an important aspect of data collection. 1

2 Sampling (Statistics) Random Sampling Random sampling is a sampling technique where we select a group of subjects (a sample) for study from a larger group (a population). Each individual is chosen entirely by chance and each member of the population has a known, but possibly non-equal, chance of being included in the sample. Simple Random Sampling Simple random sampling is the basic sampling technique where we select a group of subjects (a sample) for study from a larger group (a population). Each individual is chosen entirely by chance and each member of the population has an equal chance of being included in the sample; i.e. each member of the population is equally likely to be chosen at any stage in the sampling process. Sampling (Statistics) Simple Random Sampling Simple random sampling refers to a sampling method that has the following properties: * The population consists of N objects. * The sample consists of n objects. * All possible samples of n objects are equally likely to occur. The main benefit of simple random sampling is that it guarantees that the sample chosen is representative of the population. This ensures that the statistical conclusions will be valid. 2

3 Sampling (Statistics) To understand sampling, you need to first understand a few basic definitions: The total set of observations that can be made is called the population. A sample is a subset of a population A parameter is a measurable characteristic of a population, such as a mean or standard deviation. A statistic is a measurable characteristic of a sample, such as a mean or standard deviation; i.e., a statistic is any function of the observations in a random sample. A sampling method is a procedure for selecting sample elements from a population. The Mode Measure of Central Tendency The mode is the value that occurs the most frequently in the population or sample. Suppose we draw a sample of five women and measure their weights. They weigh 100, 100, 130, 140, and 150. Since more women weigh 100 than any other weight, the mode would equal 100 pounds. The mode is not necessarily unique, since the same maximum frequency may be attained at different values. Given the list of data [1, 1, 2, 4, 4] the mode is not unique - the dataset may be said to be bimodal, while a set with more than two modes may be described as multimodal. 3

4 Measure of Central Tendency The Median To find the median, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values. Thus, in the sample of five women, the median value would be 130 pounds; since 130 pounds is the middle weight. Measure of Central Tendency The Mean The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations. Returning to the example of the five women, the mean weight would equal ( )/5 = 620/5 = 124 pounds. 4

5 Measure of Central Tendency Proportions and Percentages When the focus is on the degree to which a population possesses a particular attribute, the measure of interest is a percentage or a proportion. A proportion refers to the fraction of the total that possesses a certain attribute. For example, we might ask what proportion of women in our sample weigh less than 135 pounds. Since 3 women weigh less than 135 pounds, the proportion would be 3/5 or A percentage is another way of expressing a proportion. A percentage is equal to the proportion times 100. In our example of the five women, the percent of the total who weigh less than 135 pounds would be 100 * (3/5) or 60 percent. Measure of Central Tendency Notation Of the various measures, the mean and the proportion are most important. The notation used to describe these measures appears below: - - x: Refers to a sample mean. n: Number of observations in the sample p: The proportion of elements in the sample that has a particular attribute. q: The proportion of elements in the sample that does not have a specified attribute. Note that q = 1 - p. X: µ: Refers to a population mean. N: Number of observations in the population P: The proportion of elements in the population that has a particular attribute. Q: The proportion of elements in the population that does not have a specified attribute. Note that Q = 1 - P. Note that capital letters refer to population parameters, and lower-case letters refer to sample statistics. 5

6 Measure of Variability Some parameters attempt to describe the amount of variation between random variables. For example, consider a population of four random variables {5, 5,5, 5}. Here, each of the random variables are equal, so there is no variation. The set {3, 5, 5, 7}, on the other hand, has some variation since some random variables are different. Measure of Variability The Range The range is the simplest measure of variation. It is difference between the biggest and smallest random variable. Range = r = Maximum value - Minimum value Therefore, the range of the four random variables (3, 5, 5, 7} would be 7 3 = 4. 6

7 Measure of Variability Variance It is important to distinguish between the variance of a population and the variance of a sample. They have different notation, and they are computed differently. s 2 : The variance of the sample. s: The standard deviation of the sample. σ 2 : The variance of the population. σ: The standard deviation of the population. Variance Measure of Variability ou may have noticed that the denominator in the calculation of sample variance, unlike the denominator in the calculation of population variance, is n and not (n-1). Why? This is because to calculate the sample variance, deviations with respect to the sample mean are used. Sample observations, tend to be closer to the sample mean than to µ. Thus, the calculated deviations are smaller. As a result, the sample variance obtained is smaller than the population variance. To compensate for this, (n -1) is used as the denominator in place of n in the calculation of sample variance. 7

8 Measure of Variability Variance : Example 1 A population consists of four observations: {1, 3, 5, 7}. What is the variance? Solution: First, we need to compute the population mean. μ = ( ) / 4 = 4, Then, σ 2 = Σ ( X i -μ ) 2 / N = [ ] / 4 = 5 Variance : Example 2 A sample consists of four observations: {1, 3, 5, 7}. What is the variance? Solution: This problem is handled exactly like the previous problem, except that we use the formula for calculating sample variance, rather than the formula for calculating population variance. s 2 = Σ ( x i - x ) 2 / ( n - 1 ) = [ ] / 3 = 20 / 3 = Measure of Variability Variance A more efficient computational formula for the sample variance is obtained as follows: = 8

9 Sampling Distributions For any given population of size N it is possible to get different samples of size n. Each sample may well have a different mean. In fact it is possible to get an entire distribution of different sample means from the various possible samples. The list of all possible values for a statistic and the probability associated with each value is known as a sampling distribution. Sampling Distributions Theorem When the population we are sampling from has, itself, roughly the shape of a normal curve, the sampling distribution of the mean can be approximated closely with a normal distribution regardless of the size of n. 9

10 For large samples ( n 30) the sampling distribution of the sample mean can be approximated closely with a normal distribution with mean equal to the population mean and standard error equal to where σ µ µ σ is the standard deviation of the population. The sampling distribution of the mean will be approximately normally distributed no matter what the population distribution looks like. σ n The standard error of the mean is defined as the standard deviation of the sampling distribution of the mean. Keep in mind that this theorem applies only to the mean and not other statistics. 10

11 Based on this theorem, if is the mean of random sample of size n from a population with mean σ and standard deviation. If n is large, then µ has approximately the standard normal distribution (using the conversion formula). Z = µ σ n Example I In this example, the population distribution shows five levels of a characteristic/variable that were measured in the population. Each level of the variable had the same probability of occurring. The population distribution is presented below. The scores in the population range from 1 to 5. The mean of the population is 3.0 and the standard deviation of the population is 1.41 [μ=3.0; σ =1.41]. 11

12 Example I Frequency distributions of sample means quickly approach the shape of a normal distribution, even if we are taking relatively few, small samples from a population that is not normally distributed (such as the uniform distribution). Example II Let's randomly select an infinite number of samples of the same size from a population that follows a Poisson distribution and calculate the mean of scores in each sample. The sampling distributions are presented as shown on the right side. 12

13 Example III Let's randomly select an infinite number of samples of the same size from a population that follows a Normal distribution and calculate the mean of scores in each sample. The sampling distributions are presented as shown on the right side. Example IV A synthetic fiber used in manufacturing carpet has tensile strength that is normally distributed with mean 75.5 psi and standard deviation 3.5 psi. Find the probability that a random sample of n = 49 fiber specimens will have sample mean tensile strength that exceeds psi. σ = μ = 75.5 psi σ = = 0.5 n µ Z = µ σ n P( 75.75) = P( Z 0.5) 13