Notes 07_Sampling Methods & CLT(1)

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CHAPTER 8: SAMPLING METHODS & CENTRAL LIMIT THEOREM

RECOMMENDED READINGCustomised Text, Adapted from ‘Statistical Techniques in Business & Economics by Lind, Marchal 16th Edition’ McGraw Hill

Chapter 7, Page 209 – 211, 218 – 240 excluding Simple Random Sampling

REVIEWProbability DistributionA probability distribution is a list of the probabilities of ALL the possible outcome of a variable in a survey / experiment.

Types of Probability Distributions(a) Population Distribution

It is the probability distribution of all values of a variable in a population.Population distributions are described by parameters.e.g. population mean, population standard deviation, and population proportion.

(b) Sample DistributionIt is the probability distribution of all values of a variable in a sample.Sample distributions are described by sample statistics.e.g. sample mean, sample standard deviation, sample proportion.Sample statistics enable us to make inferences, predictions and decisions about the parameters.

2 Sampling DistributionA sampling distribution is a probability distribution of ALL possible sample statistic values of a given sample size.

Types of Sampling Distributions Sampling distribution for the sample means Sampling distribution for the sample proportions

of 8 Sampling Distribution

Sampling distribution for sample meansA probability distribution of all possible sample mean values of a given sample size

3 Sampling distribution for sample meansIt is a probability distribution of all possible sample mean values of a given sample size.

3.1 Describing the Sampling Distribution of the Sample Mea n The two major characteristics describing the sampling distribution are:1. Mean of the sampling distributionThis is the average value of all the statistic values. (e.g. all sample mean values)

The mean of the sampling distribution for the sample means is denoted by

In general, the mean of the sampling distribution equals the population mean, i.e.

=

(Please refer to the example in pg 220 – 222 in the customised textbook.)

2. Standard Error of the sampling distributionThis is the standard deviation of all the statistic values.

The standard error for the sample mean is denoted as σ x :

Formula

σ x̄ = σ

√n if N is infinite*; or

if N is finite (given) and nN 0.05, then

σ x̄ = σ√n √ N−n

N−1

*If the population size is unknown, we assume that it is infinite.

Concepts of Standard Error

Sampling ErrorIt is unlikely that the sample mean would be identical to the population mean. We should expect some difference between the sample mean and the population. This difference is called sampling error. So,

Sampling error = The difference between each sample statistic value and the

population parameter value. i.e. |X̄ − μ |Standard Error- The average size of the sampling error in a sampling distribution.

- The standard deviation of all the sample means.

- It measures the spread of the sampling distribution for the sample means.

- It is smaller than the population standard deviation. The larger the sample size, the smaller the standard error.


3.2 Deciding the Shape of the Sampling Distribution of the Sample MeanThere are many types of distribution a sampling distribution can follow: e.g. a normal distribution, a t-distribution, f distribution or 2 distribution. For this lecture, we will learn to find probabilities using a sampling distribution that follows a normal distribution.

(a) Sampling from a Normally Distributed Population with known population standard deviationIf we sample from a population that is normally distributed with mean and standard deviation , the sampling distribution of the mean will also be normally distributed for any sample size n.

(b) Sampling from Non-Normally Distributed Population or unknown population standard deviation

Large Sample SizeIf the sample is drawn from a non-normal population, then the sampling distribution is Approximately normally distributed, is n 30. ( Central Limit Theorem applies).

Central Limit TheoremIf all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples. This is true, regardless of the shape of the distribution of the population, but if a population distribution is skewed, then it may require n ≥ 30 for the sampling distribution to approach normal distribution.

Significance of the Central Limit TheoremIt permits us to use sample statistics to make inference about the population parameters WITHOUT knowing about the shape of the population distribution.

3.3 Finding probabilities with a Sampling DistributionIn order to find probabilities from a sampling distribution of the mean that is normally distribution, we will have to convert the sample mean values to the Standard Normal value, Z.

Standard Normal Values for sampling distribution

Z =

X̄ − μx

σ x̄

= X−μσ x̄


Example 1The distribution of annual earnings of all bank tellers with five years' experience is skewed negatively with mean $15,000 and a standard deviation of $2,000. If we draw a random sample of 30 tellers, what is the probability that their earnings will average more than $15,750 annually?

Find P( X > 15750)

Step 1: Determine the standard error:

Step 2: Determine the sampling distribution and the standardized valueSince n = 30, by Central Limit Theorem, the sampling distribution is a normal distribution.

Z = = 2.05

Step 3: Sketch a diagram

Step 4: Find the probability

P( X > 15750) = P(Z > 2.05) =

Step 5: ConclusionThe probability that the mean earnings of the 30 tellers exceeding $15750 is 0.0202


Example 2Sarjit Singh, the owner of a chain of 25 clothing stores, has been thinking about winding up his business because it is not as profitable as before. Over the past few years, the mean net income for the 25 stores has been $21,000 with a standard deviation of $3,399.29. He would sell the business if the average of the first 5 stores audited at year end is less than $20,000. What is the probability that Sarjit will sell out? (Assume that the profits follow a normal distribution).

X: = annual profits of the clothing stores

Find P( X < 20,000)

Step 1: Determine the standard error:

nN

= 525

=0. 2>0 . 05 _____________ finite multiplier

σ X =

Step 2: Determine the Sampling distribution and the standardized value.The distribution of the profits of the clothing stores is a normal distribution (population distribution is normal) and the population standard deviation is known. Thus, the sampling distribution follows a normal distribution even though n < 30.

Z=20000−210001387 .75

=−0.7206

Step 3: Sketch a diagram

Step 4: Find the probability

P( X < 20000) = P(Z< -0.72) =

Step 5: Conclusion

The probability that Sarjit will sell out is 0.2358


Example 3

FEBRUARY 2010 EXAMINATION, QUESTION 3

a) Mr. Lee owns a fleet of taxis. Based on past data, he knows that the expenses incurred on each taxi is normally distributed with a mean of $120 and a standard deviation of $12. (Assume that the population follows the normal distribution and sixty taxis are randomly selected.) Mr. Lee has approached you, a business consultant, to plan for next year’s expenses by finding the following: i. Expected mean expenses incurred.

ii. Standard error of the expenses incurred

iii. Based on a random sample of sixty taxis, what is the probability that the mean expenses will exceed $123.80

iv. Probability that the mean expenses will fall between $117 and $122


Question 5The mean age at which men in USA marry for the first time follows the normal distribution with a mean of 24.8 years. The standard deviation is 2.5years. For a random sample of 60 men, what is the likelihood that the age at which they were married for the first time is less than 25.1 years.

Summary

Sample Statistics xMean of sampling distribution

μx

μx=μ=∑ x

Nwhere x = value of the variable

Standard Error

If population N is infinite or

nN

<0. 05, use

σ x=σ

√n

If population N is finite andnN

≥0 .05, then use

σ x=σ√n √ N−n

N−1

Sampling distribution

Sampling distribution is a normal distribution if: sample size n 30 by Central Limit

theorem OR population distribution is a normal

distribution and is knownStandardized Formula

Z=x−μx

σ x


Standard Normal Distribution

Z .00 .01 .02 .03 .04 .05 .06 .07 .08 .090.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .03590.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .07530.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .11410.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .15170.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .18790.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .22240.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .25490.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .28520.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .31330.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .33891.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .36211.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .38301.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .40151.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .41771.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .43191.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .44411.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .45451.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .46331.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .47061.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .47672.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .48172.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .48572.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .48902.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .49162.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .49362.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .49522.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .49642.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .49742.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .49812.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .49863.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .49903.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .49933.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .49953.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .49973.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .49983.5 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .49983.6 .4998 .4998 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .49993.7 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .49993.8 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .49993.9 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000


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Notes 07_Sampling Methods & CLT(1)

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