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# Statistics lecture 7 (ch6)

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Sampling Distributions lecture

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### Statistics lecture 7 (ch6)

1. 1. 1
2. 2. OBJECTIVES• To understand concept of sampling distribution• To understand concept of sampling error• To determine the mean and std dev for the sampling distribution of a sample mean• To determine the mean and std dev for sampling distribution of a sample proportion• To calculate the probabilities related to the sample mean and the sample proportion 2
3. 3. Sampling distributions• Can be defined as the distribution of a sample statistic.• Scientific experiments are used to make inferences concerning population parameters from sample statistics.• Need to know what is the relationship between the sample statistic and its corresponding population parameter. 3
4. 4. Sampling error• Can be defined as the difference between the calculated sample statistic and population parameter.• Sampling errors occur because only some of the observations from the population are contained in the sample.• Sampling error: sample statistic – population parameter 4
5. 5. Sampling error• Size of the sampling error depends on the sample selected.• May be positive or negative.• Should be kept as small as possible.• For smaller samples the range of possible sampling errors becomes larger.• For larger samples the range of possible sampling errors becomes smaller. 5
6. 6. CONCEPT QUESTIONS• P201 QUESTIONS 1-4 6
7. 7. Sampling distribution of the mean• Sample mean is often used to estimate the population mean.• Sampling distribution of the mean is the distribution of sample means obtained if all possible samples of the same size are selected form the population. 7
8. 8. Sampling distribution of the mean• If we calculate the average of all the sample means, say we have m such samples, the result will be the population mean: m x i x  i 1  m• The standard deviation of all the sample means, will be:  x  n referred to as the standard error of the mean 8
9. 9. Central Limit Theorem• If the sample size becomes larger, regardless of the distribution of the population from which the sample was taken, the distribution of the sample mean is approximately normally distributed: – with  x    x  – and standard deviation n• The accuracy of this approximation increases as the size of the sample increases.• A sample of at least 30 is considered large enough for the normal approximation to be applied. 9
10. 10. Properties of the sampling distribution ofthe sample mean• For a random sample of size n from a population with mean μ and standard deviation σ, the sampling distribution of x has: – a mean  x    – and a standard deviation  x  n 10
11. 11. Properties of the sampling distribution ofthe sample mean• If the population has a normal distribution, the sampling distribution of x will be normally distributed, regardless the sample size.• If the population distribution is not normal, the sampling distribution of x will be approximately normally distributed, if the sample size ≥ 30.• X N ;  2     n  11
12. 12. Example• Marks for a semester test is normally distributed, with a mean of 60 and a standard deviation of 8. – X ~ N(60;82)• A sample of 25 students is randomly selected:  – X N  x ; x 2   2   82  X N ;   N  60;   n   25  12
13. 13. Example• If we need to determine the probability that the average mark for the 25 students will be between 58 and 63. P (58  X  63)    58  60 63  60   P Z   8 8     25 25   P  1, 25  Z  1,88   0,9699  0,8944  1  0,8643
14. 14. INDIVIDUAL EXERCISE 14
15. 15. INDIVIDUAL EXERCISEThe past sales record for ice cream indicatesthe sales are right skewed, with thepopulation mean of R13.50 per customerand a std dev of R6.50. A random sample of100 sales records is selected. Find theprobability of:-1. Getting a mean of less than R13.252. Getting a mean of greater than R14.503. Getting a mean of between R13.80 and R15.20 15
16. 16. SolutionP205 - 207 of textbook 16
17. 17. WHICH EQUATION TO USE? 17
18. 18. Sampling distribution of proportion• Categorical values such as number of drivers that wear safety belts in Gauteng or number of drivers who do not wear safety belts 18
19. 19. Sampling distribution of the proportion• Population proportion will be represented by p, and the sample proportion by p  X / n, where X is ˆ the number of items with the characteristic and n is the sample size.• The standard error of the proportion is given as: p(1  p) p  n 19
20. 20. Example• Suppose that in a class of 100, 28 students fail a test.• The population proportion of students who fail the test is: X 28 p  ˆ  0, 28 n 100 20
21. 21. Example• A sample of 50 students is randomly chosen• What is the probability that more than 25% will fail the test? ˆ P P  0, 25     0, 25  0, 28   PZ    0, 28(1  0, 28)     50   P  Z  0, 47   1  0, 6808  0,3192 21
22. 22. Individual exercise/homework• Read pages 195 – 211• Self review test p 209• Supplementary exercises p209• Go to www.jillmitchell.net and view the following:-• Video on sampling distributions• Video on example of sampling distribution• Video on central limit theorem• Completely re-do the NUBE test using your textbook to assist you. 22