Sampling Distributions

Sampling Distributions - Transcript

Chapter 9 Sampling Distributions
It has been proved beyond a shadow of a doubt that smoking is one of the leading causes of statistics Fletcher Knebel


9 1 Sampling Distributions pp 456 469


Samples








Examined in order to come to a reasonable conclusion about the population from which the sample is chosen To glean meaningful information one must be statistically literate Must have an awareness of what the sample results tell us and don t tell us A statistic calculated from a sample may suffer from bias or high variability


Does not represent a good estimate of a population parameter

Vocabulary Review






Parameter an index that is related to a population Statistic an index that is related to a sample Sampling distribution of a statistic the distribution of values of a statistic taken from all possible samples of a specific size A statistic is unbiased if the mean of the sampling distribution is equal to the true value of the parameter being estimated

Rules to Review


Variance formula for a POPULATION

2


xi
N

2



Variance formula for a SAMPLE

2 s



xi x N
2



Consider the 3 element population

P 1 2 3

2 N 3 0 81649658 0 6666667
2

xi 1 2 3

x x
i

2

1 0 1 2 2 3

These values are parameters since they are derived from a population

Now consider all possible samples of size 2 with replacement


There would be 9 samples

3 9
2


Order is important

Complete the Chart
Sample Sample Mean Sample Variance Sample S D 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 MEANS

Completed Chart
Sample Sample Mean Sample Variance Sample S D

1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 MEANS

1 1 5 2 1 5 2 2 5 2 2 5 3 2

0 25 1 25 0 25 1 25 0 33

0 1 5 5 0 5 1 5 0

The chart shows that






The mean of the distribution of sample means is the mean of the population This illustrates that a sample mean is an unbiased estimator of the population mean The distribution of sample means centers around the mean of the population







The mean of the distribution of sample variances s2 is equal to the variance 2 of the population This illustrates that a sample variance s2 is an unbiased estimator of the population variance The distribution of sample variances centers around the variance of the population

Take Note






A sample standard deviation is NOT an unbiased estimator of the population standard deviation In the above example the mean of the sample deviation is 0 628539 and the standard deviation for the population is 8 81649658 The distribution of sample standard deviations does not center around the standard deviation of the population

9 2 Sample Proportions pp 472 477 The normal distribution curve is often
extremely useful in analyzing sample proportions This section provides insights into the circumstances that allow for use of normal distribution properties

Consider an SRS of 1000 people from a large population


X represents the number in this sample who are Republicans


There are 1001 possible values for X 0 1 1000



p represents the possible sample proportions of P hat Republicans in the sample





There are 1001 possible values of p hat 0 1000 1 1000 1000 1000



We could choose many SRS s and calculate a p hat for each We would expect the distribution of p hat to be approximately normal

If we choose an SRS of size n from a large population with population proportion p having some characteristic of interest and if p hat is the proportion of the sample having that characteristic then






The sampling distribution of p hat is approximately normal The mean of the sampling distribution is p the population parameter The standard deviation of the sampling distribution is p 1 p

n

It is reasonable to use the previous statements when


The population is at least 10 times as large as the sample


Rule of Thumb 1



Np is at least 10 and n 1 p is at least 10


Rule of Thumb 2

Suppose it is known that 60 of the registered voters in a district of over 20 000 people are Republicans IF YOU CHOOSE AN SRS OF 1000 REGISTERED VOTERS


What is the probability that the proportion of registered voters in the sample is between 58 and 62 What is the probability that the sample will contain more than 550 Republicans Are both rules of thumb satisfied







Convert x 55 to its z score Interpret Rare occurrence 000628 is approx 1 1592 So if we had 1600 random samples of size 1000 how many of them would we expect to have 550 or fewer Republicans







9 3 Sample Means pp 481 494




This section contains one of the most important of all statistical theorems the Central Limit Theorem of Statistics It also emphasizes that it is conventionally the Greek letters and that are used for the population parameters mean and standard deviation and x bar and s are used to represent the mean and standard deviation for samples

The Central Limit Theorem


Consider an SRS of size n from any population with mean and standard deviation When n is large the sampling distribution of x bar has the following properties


It is approximately normal The mean of the distribution is x bar The standard deviation of the distribution is s n

Consider the population P 2 4 6
4

2 4

2

4 4 6 4 1 632993162 n
2 2



Now consider all possible sample size 2 with NO REPLACEMENT There would be 3x3 or 9 such samples

Sample
2 2 2 4 2 6 4 2 4 4 4 6 6 2 6 4 6 6

Sample Mean
2 3 4 3 4 5 4 5 6



The mean of the sample means is equal to 4 which is equal to


This illustrates the second part of the Central Limit Theorem



The standard deviation of the sample means is equal to 1 154700538

1 632993162 2 2


This illustrates the third part of the Central Limit Theorem