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# Sampling

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### Sampling

1. 1. Sample and Sampling Procedure <ul><li>Population </li></ul><ul><ul><li>The income of all families living in a city. </li></ul></ul><ul><ul><li>The annual yields of all stocks traded on a stock exchange </li></ul></ul><ul><ul><li>The costs of all claims for automobile accident insurance coverage received by a co. in a given year. </li></ul></ul><ul><ul><li>The mileage achieved by all cars of a particular model </li></ul></ul><ul><li>We might be interested in learning about specific characteristics, or attributes of these populations. For example, we might want to make an inference about the mean or variance of the population distribution of family incomes in the city, or about the proportion of all families in the city with annual incomes below Rs 12000. </li></ul><ul><li>The major motivation for examining a sample rather than the whole population is that the collection of the complete information on the latter would be difficult and typically expensive. </li></ul><ul><li>Larger Parent Group : Population . </li></ul><ul><li>Subset : Sample . </li></ul><ul><li>If a sample is to be taken from a population, the eventual aim is to make statements that have some validity for the population at large. It is important that the sample be a representative of the population. </li></ul>
2. 2. Randomness in Sampling <ul><li>Suppose a marketing manager wants to assess the reactions to a new food product. </li></ul><ul><li>It would be unwise of him to restrict his survey to his circle of friends or to people living in his immediate neighborhood. Such groups are very unlikely to reflect the spectrum of views of the population at large. </li></ul><ul><li>It is important that the principle of randomness to be embodied in the sample selection procedure. That is every sample of same size has the equal chance of selection. </li></ul>
3. 3. Random sampling <ul><li>Suppose it is required to select a sample of “n” objects from a population of “N” objects. A random sampling procedure is one in which every possible sample of n objects is equally likely to be chosen. The resulting sample is called a random sample. </li></ul><ul><li>This type of sampling and the resulting sample makes a true representation of the population. </li></ul>
4. 4. Sampling Distribution <ul><li>On the basis of sample information our objective is to make inference about the whole population. </li></ul><ul><li>Ex. Given a random sample of mileage/liter achieved by 20 cars of a particular model, one can make inferential statements about the mean and variance of mileage of all cars of that model. </li></ul><ul><li>Since from sample to sample, different values will result for the sample mean, this quantity is regarded as a random variable, which has a probability distribution known as sampling distribution (of means). </li></ul>
5. 5. Sampling Distribution <ul><li>Statistics. </li></ul><ul><ul><li>A particular function of the sample (Mean, Variance, Median etc). </li></ul></ul><ul><li>The sampling distribution of this statistics is the probability distribution of the values it could take over all possible samples of the same number of observations drawn from the population. </li></ul>
6. 6. Example <ul><li>Let us the position of the supervisor with 6 employees whose experiences are </li></ul><ul><li>2, 4, 6, 6 , 7, 8 (mean=5.5) </li></ul><ul><li>Four of these employees are to be chosen randomly and assigned a particular work shift. </li></ul><ul><li>There are 15 possible samples that could be selected and they may have max 15 means. </li></ul>
7. 7. Sample Means -Probability
8. 8. Sampling Distribution Mean <ul><li>The mean of the sampling distribution of the means is the population mean. </li></ul><ul><li>If the samples of n observations are repeatedly and independently drawn from a population, than as the numbers of samples are taken becomes very large the average of the sample means becomes very close to the true population mean. </li></ul><ul><li>In the above example the mean of the sample means is equal to 5.5 (the population mean also). </li></ul>
9. 9. Sampling Distribution of the sample mean <ul><li>Let X 1 , X 2 , …,X n be a random sample from a population. The average value: </li></ul>Of these observations is called the sample mean. We need to consider the sampling distribution of the random variable X.
10. 10. Variance of Sampling Distribution of Means <ul><li>Standard error = </li></ul><ul><li>This says that the more observation in the sample the concentrated is the sampling distribution of the sample mean about the population mean. </li></ul><ul><li>If the number of sample member n is not very small fractions of the number population members N . In that case </li></ul>
11. 11. Sampling mean distribution from a normal population <ul><li>If the parent population is normal, it is possible to show that the sample means are also normally distributed. Thus in the standard normal form we have : </li></ul>Moreover, by Central Limit theorem, even if the population distribution is not normal, but the sample size is moderately large, then the distribution of X, will still be very close to normal.
12. 12. Examples-1 and 2. <ul><li>If the number of miles per gallon achieved by all cars of a particular model has mean 25 and SD 2, what is the probability that, for a random sample of 20 such cars, average miles per gallon is will be less than 24 ? Assume that the population distribution is normal. </li></ul><ul><li>A manufacturer claims that the life of its spark plugs is N distributed with mean 36,000 miles and SD 4000 miles. For a random sample of 16 of these plugs, the average life was found to be 34500 miles. If the manufacturer’s claim is correct, what would the probability of finding a sample mean this small, or smaller ? </li></ul>
13. 13. Sample Proportion Distribution <ul><li>Let X be a number of successes in a binomial sample of n observations , where the probability of success is p ( In most applications the parameter p is the proportion of members of a large population possessing a characteristic of interest). Then the proportion of success </li></ul><ul><li>In the sample is called sample proportion </li></ul>
14. 14. Sampling distribution of the Sample proportion.
15. 15. <ul><li>Notice that, for fixed p , the standard error of the sample proportion decreases as the sample size increases. That is, for increasing sample size, the distribution of the becomes more concentrated about its mean. </li></ul><ul><li>In other words if we take a bigger sample from the population, our inference about the proportion of population members possessing some particular characteristic becomes more firm. </li></ul>
16. 16. Examples <ul><li>Ex-3. A corporation receives 100 applications for a position from recent college graduates in business. Assuming that these applicants can be regarded as the random sample of all graduates, what is the probability that between 25% to 35% of them are women if 30% of all recent graduates in business are women. </li></ul>
17. 17. Examples <ul><li>Ex-4. A random sample of 205 students in adult education classes on a large college campus it was found that 66% of the sample members were not in the market for cigarettes. What would be the probability of observing a sample proportion as high or higher than this if in fact 60% of all adult education students on the campus are not in the market for cigarettes. </li></ul>
18. 18. Sampling distribution of the sample variance <ul><li>Let X1,X2, …,Xn be a random sample from a population. The quantity </li></ul>Is called the sample variance. Its square root is called the sample standard deviation
19. 19. Chi-Square Distribution <ul><li>If the population is normal , we can have the  2 distribution with (n-1) degrees of freedom as given below. </li></ul>The mean and variance of  2 distribution are equal to the number of degrees of freedom and twice the number of degrees of freedom respectively .
20. 20. Sampling distribution of the sample variance <ul><li>Degree of freedom is n-1 </li></ul>
21. 21. Example <ul><li>Ex-4. A manufacturer of canned peas is concerned with the mean weight of the product be close to the advertised weight . In addition he does not want too much variability in the weights of the canned peas ; otherwise a large proportion will differ markedly from the advertised weight. Assume that the population distribution of the weights is normal. If a random sample of 20 cans is checked, find the numbers K1 and K2 such that </li></ul>