SAMPLING DISTRIBUTION AND POINT ESTIMATION OF PARAMETERS - Copy.pptx

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1. SAMPLING DISTRIBUTION AND POINT ESTIMATION OF PARAMETERS

2. Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population . These methods utilize the information contained in a sample from the population in drawing conclusions. Statistical inference may be divided into two major areas: Parameter estimation Hypothesis testing

3. ESTIMATION A process whereby we select a random sample from a population and use a sample statistic to estimate a population parameter .

4. EXAMPLES OF ESTIMATION an auto company may want to estimate the mean fuel consumption for a particular model of a car a manager may want to estimate the average time taken by new employees to learn a job; the U.S. Census Bureau may want to find the mean housing expenditure per month incurred by households; and the AWAH ( Association of Wives of Alcoholic Husbands) may want to find the proportion (or percentage) of all husbands who are alcoholic.

5. Take a subset of the population Estimations Lead to Inferences

6. Estimations Lead to Inferences Try and reach conclusions about the population

7. The estimation procedure involves the following steps : Select a sample. Collect the required information from the members of the sample. Calculate the value of the sample statistic. Assign value(s) to the corresponding population parameter.

8. Inferential Statistics involves Three Distributions: A population distribution – variation in the larger group that we want to know about. A distribution of sample observations – variation in the sample that we can observe. A sampling distribution – a normal distribution whose mean and standard deviation are unbiased estimates of the parameters and allows one to infer the parameters from the statistics.

9. ESTIMATION

10. ESTIMATION

11. POINT ESTIMATION A sample statistic is used to estimate the exact value of a population parameter.

12. Point Estimator A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or a point.

13. Sampling Distributions and the Central Limit Theorem Statistical inference is concerned with making decisions about a population based on the information contained in a random sample from that population. Definitions:

14. THE CENTRAL LIMIT THEOREM As the sample size n increases without limit, the shape of the distribution of the sample means taken with replacement from a population with mean µ and standard deviation σ will approach a normal distribution. As previously shown, this distribution will have a mean µ and a standard deviation σ / √n .

15. 7.2 Sampling Distributions and the Central Limit Theorem Figure 7-1 Distributions of average scores from throwing dice. [Adapted with permission from Box, Hunter, and Hunter ( 1978). ]

16. Sampling Distributions and the Central Limit Theorem

17. Procedure Table Draw a normal curve and shade the desired area . Convert the X value to a z value . Find the corresponding area for the z value.

18. Sampling Distributions and the Central Limit Theorem It’s important to remember two things when you use the central limit theorem: When the original variable is normally distributed, the distribution of the sample means will be normally distributed, for any sample size n. When the distribution of the original variable is not normal, a sample size of 30 or more is needed to use a normal distribution to approximate the distribution of the sample means. The larger the sample, the better the approximation will be.

19. Sampling Distributions and the Central Limit Theorem 1. An electronics company manufactures resistors that have a mean resistance of 100 ohms and a standard deviation of 10 ohms. The distribution of resistance is normal. Find the probability that a random sample of n = 25 resistors will have an average resistance less than 95 ohms.

20. Sampling Distributions and the Central Limit Theorem 2. A . C. Neilsen reported that children between the ages of 2 and 5 watch an average of 25 hours of television per week. Assume the variable is normally distributed and the standard deviation is 3 hours. If 20 children between the ages of 2 and 5 are randomly selected , find the probability that the mean of the number of hours they watch television will be greater than 26.3 hours.

21. Sampling Distributions and the Central Limit Theorem 3. The average time spent by construction workers who work on weekends is 7.93 hours (over 2 days). Assume the distribution is approximately normal and has a standard deviation of 0.8 hour. If a sample of 40 construction workers is randomly selected, find the probability that the mean of the sample will be less than 8 hours.

22. Sampling Distributions and the Central Limit Theorem 4. The average age of a vehicle registered in the United States is 8 years, or 96 months. Assume the standard deviation is 16 months. If a random sample of 36 vehicles is selected , find the probability that the mean of their age is between 90 and 100 months.

23. Sampling Distributions and the Central Limit Theorem Approximate Sampling Distribution of a Difference in Sample Means

24. General Concepts of Point Estimation: Unbiased Estimators

25. What can I do to Ensure Unbiasedness in my Data or Sampling Distribution? Take your  sample  according to sound statistical practices. Avoid   measurement error  by making sure data is collected with unbiased practices. For example, make sure any questions posed aren’t ambiguous. Avoid unrepresentative samples by making sure you haven’t excluded certain population members (like minorities or people who work two jobs).

26. General Concepts of Point Estimation: Variance of a Point Estimator Figure 7-5 The sampling distributions of two unbiased estimators

27. General Concepts of Point Estimation: Variance of a Point Estimator

28. General Concepts of Point Estimation: Standard Error: Reporting a Point Estimate

29. General Concepts of Point Estimation: Standard Error: Reporting a Point Estimate

30. General Concepts of Point Estimation: Mean Square Error of an Estimator

31. General Concepts of Point Estimation: Mean Square Error of an Estimator

32. General Concepts of Point Estimation: Mean Square Error of an Estimator Figure 7-6 A biased estimator that has smaller variance than the unbiased estimator

33. Three Properties of a Good Estimator The estimator should be an unbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated. The estimator should be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated. The estimator should be a relatively efficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.