STA 291 Fall 2009

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STA 291 Fall 2009. Lecture 12 Dustin Lueker. Reduce Sampling Variability. The larger the sample size, the smaller the sampling variability Increasing the sample size to 25…. 10 samples of size n=25. 100 samples of size n=25. 1000 samples of size n=25. Interpretation.

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### STA 291Fall 2009

Lecture 12

Dustin Lueker

Reduce Sampling Variability
• The larger the sample size, the smaller the sampling variability
• Increasing the sample size to 25…

10 samples

of size n=25

100 samples

of size n=25

1000 samples

of size n=25

STA 291 Fall 2009 Lecture 12

Interpretation
• If you take samples of size n=4, it may happen that nobody in the sample is in AS/BE
• If you take larger samples (n=25), it is highly unlikely that nobody in the sample is in AS/BE
• The sampling distribution is more concentrated around its mean
• The mean of the sampling distribution is the population mean

STA 291 Fall 2009 Lecture 12

Sampling Distribution

• If you repeatedly take random samples and

calculate the sample mean each time, the

distribution of the sample mean follows a

pattern

• This pattern is the sampling distribution

Population with mean m and standard deviation s

STA 291 Fall 2009 Lecture 12

Example of Sampling Distribution of the Mean

As n increases, the variability decreases and the normality (bell-shapedness) increases.

STA 291 Fall 2009 Lecture 12

Effect of Sample Size
• The larger the sample size n, the smaller the standard deviation of the sampling distribution for the sample mean
• Larger sample size = better precision
• As the sample size grows, the sampling distribution of the sample mean approaches a normal distribution
• Usually, for about n=30, the sampling distribution is close to normal
• This is called the “Central Limit Theorem”

STA 291 Fall 2009 Lecture 12

Examples
• If X is a random variable from a normal population with a mean of 20, which of these would we expect to be greater? Why?
• P(15<X<25)
• P(15< <25)
• What about these two?
• P(X<10)
• P( <10)

STA 291 Fall 2009 Lecture 12

Mean of sampling distribution
• Mean/center of the sampling distribution for sample mean/sample proportion is always the same for all n, and is equal to the population mean/proportion.

STA 291 Fall 2009 Lecture 12

Reduce Sampling Variability
• The larger the sample size n, the smaller the variability of the sampling distribution
• Standard Error
• Standard deviation of the sample mean or sample proportion
• Standard deviation of the population divided by

STA 291 Fall 2009 Lecture 12

Sampling Distribution of the Sample Mean
• When we calculate the sample mean, , we do not know how close it is to the population mean
• Because is unknown, in most cases.
• On the other hand, if n is large, ought to be close to

STA 291 Fall 2009 Lecture 12

Parameters of the Sampling Distribution
• If we take random samples of size n from a population with population mean and population standard deviation , then the sampling distribution of
• has mean
• and standard error
• The standard deviation of the sampling distribution of the mean is called “standard error” to distinguish it from the population standard deviation

STA 291 Fall 2009 Lecture 12

Standard Error
• The example regarding students in STA 291
• For a sample of size n=4, the standard error of is
• For a sample of size n=25,

STA 291 Fall 2009 Lecture 12

Central Limit Theorem
• For random sampling, as the sample size n grows, the sampling distribution of the sample mean, , approaches a normal distribution
• Amazing: This is the case even if the population distribution is discrete or highly skewed
• Central Limit Theorem can be proved mathematically
• Usually, the sampling distribution of is approximately normal for n≥30
• We know the parameters of the sampling distribution

STA 291 Fall 2009 Lecture 12

Example
• Household size in the United States (1995) has a mean of 2.6 and a standard deviation of 1.5
• For a sample of 225 homes, find the probability that the sample mean household size falls within 0.1 of the population mean
• Also find

STA 291 Fall 2009 Lecture 12

Sampling Distribution

• If you repeatedly take random samples and

calculate the sample proportion each time, the

distribution of the sample proportion follows a

pattern

Binomial Population with proportion p of successes

STA 291 Fall 2009 Lecture 12

Example of Sampling Distributionof the Sample Proportion

As n increases, the variability decreases and the normality (bell-shapedness) increases.

STA 291 Fall 2009 Lecture 12

Central Limit Theorem (Binomial Version)
• For random sampling, as the sample size n grows, the sampling distribution of the sample proportion, , approaches a normal distribution
• Usually, the sampling distribution of is approximately normal for np≥5, nq≥5
• We know the parameters of the sampling distribution

STA 291 Fall 2009 Lecture 12

Example
• Take a SRS with n=100 from a binomial population with p=.7, let X = number of successes in the sample
• Find
• Does this answer make sense?
• Also Find
• Does this answer make sense?

STA 291 Fall 2009 Lecture 12