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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

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