CABT SHS Statistics & Probability - Mean and Variance of Sampling Distributions of Sample Means

Sigma/Summation Notation

The session shall begin shortlyCABT Statistics & Probability Grade 11 Lecture Presentation

(groupie muna)

1

LETS START!

Mean & Variance of Sampling Distributions of Sample MeansA Grade 11 Statistics and Probability Lecture

At the end of this lesson, you are expected to: find the mean and variance of the sampling distribution of the sample means. state and explain the Central Limit Theorem. use the Central Limit Theorem to solve problems involving means and variances of sampling distribution of means.Lesson ObjectivesMean and Variance of Sampling Distributions of Sample Means

Review: Finding the Mean and Variance of Discrete Probability DistributionsCABT Statistics & Probability Grade 11 Lecture PresentationMean and Variance of Sampling Distributions of Sample Means

XP(X)10.120.330.240.4

X P(X)0.10.60.61.6

X2P(X)0.11.21.86.4

2.9

9.5

SUMS

Mean:

Variance:

Standard deviation:

5

Recall: Sampling Distribution of the MeanCABT Statistics & Probability Grade 11 Lecture PresentationA sampling distribution of sample means is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population.The probability distribution of the sample means is also called the sampling distribution of the sample means, with the sample mean as the random variable.

http://www.philender.com/courses/intro/notes2/sam.gif

Mean and Variance of Sampling Distributions of Sample Means

6

Recall: Constructing Sampling Distribution of Sample MeansCABT Statistics & Probability Grade 11 Lecture Presentation

Mean and Variance of Sampling Distributions of Sample MeansHow to Construct a Sampling Distribution of Sample Means from a Given PopulationSTEP 1 Determine the number of samples of size n from the population of size N.STEP 2 List all the possible samples and compute the mean of each sample.STEP 3 Construct a frequency distribution of the sample means obtained in Step 2.STEP 4 Construct the probability distribution P(X) for the sample means X.

7

CABT Statistics & Probability Grade 11 Lecture Presentation

Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample MeansBecause the sampling distribution P(X) ofsample means X is essentially a discrete probability distribution, the computation of the mean and variance of P(X) is the SAME as for any discrete probability distribution P(X).

Ows, talaga?Mismo!

8


Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample MeansFor the sampling distribution of the sample means :

Mean:

Variance:

Standard deviation:

9


Consider a group of N = 4 people with the following ages: 18, 20, 22, 24. Consider samples of size n = 2 from the group without replacement. If X is the age of one person in the group and X is the average age of the two people in a sample, find the mean and variance of the sampling distribution of X. Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample Means

1

10


Sampling without replacementNumber of samples of size 2:

SampleSample Mean18, 201918, 222018, 242120, 222120, 24 2222, 2423

Sample MeanFrequencyProbability1911/62011/62121/32211/62311/6TOTAL61

Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample Means

1

Solution

11


Sample MeanFrequencyProbability1911/62011/62121/32211/62311/6TOTAL61

XP(X)191/6201/6211/3221/6231/6

Mean and Variance of Sampling Distributions of Sample MeansSampling without replacementNumber of samples of size 2:

1

SolutionThe Mean and Variance of Sampling Distribution of Sample Means

12


Computing the mean and variance:Mean and Variance of Sampling Distributions of Sample MeansXP(X)191/6201/6211/3221/6231/6

X P(X)X2 P(X)3.1760.173.3366.677.00147.003.6780.673.8388.17

21.00

442.67

SUMS

Mean:

Variance:

1

SolutionThe Mean and Variance of Sampling Distribution of Sample Means

13


Redo Example 1 if sampling is with replacement.Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample Means

Sampling with replacementNumber of samples of size 2:

2

Solution

2

SampleMeanSampleMeanSampleMeanSampleMean18,181820, 181922,182024,182118, 201920, 202022, 202124, 202218, 222020, 222122, 222224, 222318, 242120, 242222, 242324, 2424

14


Mean and Variance of Sampling Distributions of Sample MeansSampling with replacementNumber of samples of size 2:

2

SolutionSample MeanFrequencyProbability1811/161921/82033/162141/42233/162321/82411/16TOTAL161

XP(X)181/16191/8203/16211/4223/16231/8241/16

The Mean and Variance of Sampling Distribution of Sample Means

15



XP(X)181/16191/8203/16211/4223/16231/8241/16

Computing the mean and variance:X P(X)X2 P(X)1.1320.252.3845.133.7575.005.25110.254.1390.752.8866.131.5036.00

21

443.5

SUMS

Mean:

Variance:

16



Consider a population with values1, 1, 2, 3. Samples of size 3 are drawn from the population without replacement.Construct a sampling distribution of the sample means for this population.Find the mean, variance, and standard deviation of the sampling distribution.SOLUTION HERE!The Mean and Variance of Sampling Distribution of Sample Means

3

17

Do you haveany QUESTIONs?

Intermission Number


Mean and Variance of Sampling Distributions of Sample MeansQUESTION!

How are the ACTUAL mean and variance of the population related to the mean and variance of the sampling distribution?The Mean and Variance of Sampling Distribution of Sample Means

20


Example 1 and 2 revisited: Population: (18, 20, 22, 24)Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample Means

Mean of the population:

XX - (X - )21839201122 1124 39sum20

Variance of the population:

21


Lets compare! Mean and Variance of Sampling Distributions of Sample MeansMeanVariancePopulationSampling Distribution(samples of size 2 without replacement)

Population: (18, 20, 22, 24)Sampling: n = 2, without replacementThe Mean and Variance of Sampling Distribution of Sample Means

22


Lets compare! Mean and Variance of Sampling Distributions of Sample MeansMeanVariancePopulationSampling Distribution(samples of size 2 with replacement)

Population: (18, 20, 22, 24)Sampling: n = 2, with replacementThe Mean and Variance of Sampling Distribution of Sample Means

23


Lets compare! Mean and Variance of Sampling Distributions of Sample MeansValues of variances (N = 4, n = 2)The Mean and Variance of Sampling Distribution of Sample Means

Without replacement:

With replacement:

24


Mean and Variance of Sampling Distributions of Sample MeansProperties of the Mean and Variance of Sampling Distributions of Sample Means


ConditionMEANVARIANCEFor infinite populations or for sampling with replacementFor finite populations or for sampling without replacement

25


Mean and Variance of Sampling Distributions of Sample MeansProperties of the Mean and Variance of Sampling Distributions of Sample Means


The mean of the sample means will be the same as the population mean.The variance or standard deviation of the sample means will be smaller than the variance or standard deviation of the population.

26


Mean and Variance of Sampling Distributions of Sample MeansIf a population is normal with mean and standard deviation , the sampling distribution of is also normally distributed with the following:


meanstandard deviation*variance

*Note: For , you can also use

27


Mean and Variance of Sampling Distributions of Sample MeansIf the population is infinite with mean and standard deviation or the sampling is with replacement, use the following:




28


Mean and Variance of Sampling Distributions of Sample MeansIf the population is finite with mean and standard deviation or the sampling is without replacement, use the following:




29

Normal Population DistributionNormal Sampling Distribution (has the same mean)


For a normal distribution:

For sampling with replacement:

Largersample sizeSmaller sample size


As the sample size n increases, decreases.



The Standard Error of the MeanA measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean, which is simply equal to the standard deviation of the sampling distribution of the sample means :Note that the standard error of the mean decreases as the sample size increases.

32



The Finite PopulationCorrection Factor

The FINITE POPULATION CORRECTION FACTOR is used for computing the standard error of the mean from a finite population. It is given by the expression

Wow!

33



The Finite PopulationCorrection FactorThe correction factor is necessary if relatively large samples are taken from a small population, because the sample mean will then more accurately estimate the population mean and there will be less error in the estimation.Why the need for the correction factor?

34



The Finite PopulationCorrection FactorThe finite population correction factor is also used when the sample size n is LARGE relative to the population N. This means that n is larger than 5% of N:

For example, if N = 10 and n = 6,

so the correction factor is needed in this case.

35


Consider an infinite population with mean 20 and standard deviation 2. Samples of size 4 are obtained from the population. Determine the following:Mean of the sampling distributionVariance of the standard distributionStandard error of the mean

4


36

4

SolutionMean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample Means

CABT Statistics & Probability Grade 11 Lecture PresentationGiven: = 20, = 2, n = 4Since the population is normal and infinite, we use the following formulas: meansampling error of the mean(standard deviation)variance


Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distribution of Sample MeansConsider a finite population with size 10 with mean 20 and standard deviation 2. Samples of size 4 are obtained from the population without replacement. Determine the following:Mean of the sampling distributionVariance of the standard distributionStandard error of the mean

5

38

5

SolutionMean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distributions of Sample Means

CABT Statistics & Probability Grade 11 Lecture PresentationGiven: = 20, = 2, n = 4, N = 10Since the population is finite and sampling is without replacement, we use the following formulas: meansampling error of the mean(standard deviation)variance



If a population is normal with mean and standard deviation , the sampling distribution of the sample means is also normally distributed, regardless of the sample size.

What can we say about the shape of the sampling distribution of the sample means when the population from which the sample is selected is not normal?QUESTION!

Mean and Variance of Sampling Distributions of Sample MeansThe Mean and Variance of Sampling Distributions of Sample Means

41

The Central Limit TheoremThe World is Normal Theorem


The mean and standard deviation of the distribution are, respectively, If random samples of size n are drawn from a population with replacement, then as n becomes larger, the sampling distribution of the mean approaches the normal distribution, regardless of the shape of the population distribution. Mean and Variance of Sampling Distributions of Sample MeansThe Central Limit Theorem

43


What does thistheorem mean?

If the sample size n drawn from a finite population of any shape is LARGE enough, then it is safe to ASSUME that the distribution is APPROXIMATELY NORMAL, and thus the following can be used:

Mean and Variance of Sampling Distributions of Sample MeansThe Central Limit Theorem

44


How large islarge enough?

According to most statistics books, the sample size n is LARGE ENOUGH when

Wow!Mean and Variance of Sampling Distributions of Sample MeansThe Central Limit Theorem

45

From Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Population Distribution(not normal)Sampling Distribution (becomes normal as n increases)

Larger sample sizeSmaller sample sizeSampling distribution properties:


The Central Limit TheoremILLUSTRATION OF THE CENTRAL LIMIT THEOREMFOR THREE POPULATIONS(from Statistics for Business and Economics by Anderson, Sweeney and Williams, 10th ed)

47


Two things to remember in using 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 might not be 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.


48


RECALL: If the distribution of a random variable X is normal with mean and standard deviation , the equivalent z-scores is obtained by using

This formula transforms the values of the variable x into standard units or z values. Mean and Variance of Sampling Distributions of Sample MeansApplication of theCentral Limit Theorem

49


For a normally (or approximately normally) distributed population with mean and standard deviation , the equivalent z-score of a sample mean from the sampling distribution of means is given by the formula

orMean and Variance of Sampling Distributions of Sample MeansApplication of theCentral Limit Theorem

50


Distribution of Sample Means for a Large Number of Samples

Mean and Variance of Sampling Distributions of Sample MeansApplication of theCentral Limit Theorem

51


A population is normally distributed with mean 20 and standard deviation 6. Samples of size 9 are drawn from the population. What are the equivalent z-scores of the following sample means?

1

a. b.


52


A population is normally distributed with mean 20 and standard deviation 6. Samples of size 9 are drawn from the population. What is the probability that the sample mean isgreater than 22?between 22 and 25?

2


53


A population has mean 50 and standard deviation 12. Samples of size 36 are drawn from the population. What is the probability that the sample mean isless than 48?between 42 and 50?greater than 53.5?

3


54


A recent study of a company regarding the lifetimes of cell phones it issues to its employees revealed that the average is 24.3 months and the standard deviation is 2.6 months.

4

a. If a company phone is chosen at random, what is the probability that the phone will last for more than 3 years? b. If a company randomly samples 40 phones, what is the probability that the average lifetime of the sample is more than 3 years? Mean and Variance of Sampling Distributions of Sample MeansApplication of theCentral Limit Theorem

55


Exercise 4 on page 133, edited

5

In a study of life expectancy of 400 people in a certain geographic region, the mean age of death is 70 years and the standard deviation is 5.1 years. If a sample of 50 people from this region is selected, what is the probability that the mean life expectancy is less than 68 years?


56

CABT Statistics & Probability Grade 11 Lecture PresentationCheck your understanding

Do Exercise 2 on page 132 and Extension A#1 on p. 133 of your textbook.Mean and Variance of Sampling Distributions of Sample Means

57

CABT Statistics & Probability Grade 11 Lecture PresentationAssignmentDo Exercise #1 on p. 132, due on Thursday,August 25.Mean and Variance of Sampling Distributions of Sample Means

Quiz on the Central Limit Theorem onTuesday, September 13.

58


Summing it up!

Thank you!

CABT Statistics & Probability Grade 11 Lecture PresentationQuiz #1A population has a mean of 90 and standard deviation 3.5. Samples of size 49 are taken from the population and the sample means are taken. 1. Determine the z-scores of the following sample means: Mean and Variance of Sampling Distributions of Sample Means

2. What is the probability that the sample mean will lie between 88.5 and 90.15?

62

References

Statistics and Probability 2016 ed. Rex Book Store, Inc.Next Century Mathematics Statistics and Probability. Phoenix Publishing House, Inc.Elementary Statistics: A Step-by-Step Approach 8th ed. The McGraw-Hill Companies. Statistical Analysis with Excel for Dummies 2nd ed. Wiley Publishing, Inc.Sampling and Sampling Distributions. http://home.sandiego.edu/~kdoshi/QBA%20Powerpoints/chap07.ppt Sampling Distributions. http://www.pstat.ucsb.edu/faculty/yuedong/5E/5Enote8.pptSampling Distribution Models and the Central Limit Theorem. http://www.stat.ncsu.edu/people/reiland/courses/st311/chap17_sampling%20distributions.ppt

CABT SHS Statistics & Probability - Mean and Variance of Sampling Distributions of Sample Means

Education

Transcript of CABT SHS Statistics & Probability - Mean and Variance of Sampling Distributions of Sample Means