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### Sampling Distribution

Tripthi M. Mathew, MD, MPH

Objectives

- Learning Objective

To understand the topic on Sampling Distribution and its importance in different disciplines.

- Performance Objectives

At the end of this lecture the student will be able to:

- Apply the basic knowledge of sampling distribution to solve problems.
- Interpret the results of the problems.

Types of Distribution

- Frequency Distribution
- Normal (Gaussian) Distribution
- Probability Distribution
- Poisson Distribution
- Binomial Distribution
- Sampling Distribution
- t distribution
- F distribution

What is Sampling Distribution?

- Sampling is defined as the process of selecting a number of observations (subjects) from all the observations (subjects) from a particular group or population.
- Sampling distribution is defined as the frequency distribution of the statistic for many samples.
- It is the distribution of means and is also called the sampling distribution of the mean.

Features of Sampling Distribution

The 4 features of sampling distribution include:

1) The statistic of interest (Proportion, SD, or Mean)

2) Random selection of sample

3) Size of the random sample (very important)

4) The characteristics of the population being sampled.

Characteristics of Sampling Distribution

- Central Limit Theorem

When random samples of size is taken from a population, the distribution of sample means will approach the normal distribution.

- When the Sampling distribution of the mean has sample sizes of 30 or more then it is said to be normally distributed.

Statistical Characteristics of Sampling Distribution

The major statistics are:

Mean

Standard deviation

Standard error

- The standard error (SE or SEM) of the sampling distribution is given by the formula:

s

√n

Where, n - sample size

s- standard deviation of the sample

x – sample mean

Statistical Characteristics of Sampling Distribution Cont’d

a)SE of a proportion = √ p (1-p)/n

Where, p is the sample proportion

b)SE of a percentage=√ p (100-p)/n

Where, p is the sample percentage

Statistical Characteristics of Sampling Distribution Cont’d

Confidence Interval

a) CI = p ± z α/2 √ p (1-p)/n

b) CI= p ± z α/2 √ p (100-p)/n

Statistical Characteristics of Sampling Distribution Cont’d

Z Score (Standard Score)

Z = x- μ

σ /√n

Where, X is the sample mean

μ is the mean of the sampling distribution

σis the SE of the sampling distribution

√n

Exercises

- An Epidemiologist studied a randomly selected group of 25 individuals (men and women) between 30- 49 years of age and finds that their mean heart rate is 70 beats per minute.

Exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

Exercise # 1

- How frequently will the sample of 25 individuals have a mean heart rate of 74 beats per minute or higher?

or in other words

- What proportion of samples will have mean values of 74 beats per minute or greater, if repeated samples of 25 individuals are randomly selected from the population?

Exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

Exercise # 2

- Further investigation revealed that the 25 individuals appeared to have used a drug for treatment and now the epidemiologist (Epi) wants to detect the adverse effects of the drug on the heart rate. The Epidemiologist assumes that a mean heart rate in the upper 5% of the distribution will be cause for concern.

Determine the value that divides the upper 5% from the lower 95% of the sampling distribution.

Exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

The Use of Normal Curve to solve problemsExercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

95%

5%

73.29

μ

1

2

Exercise # 3Exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

- The “disease detective” (Epi) wants to know how many patients should be included in the study to determine the drug’s effect. The Epi assumes that the mean heart rate must not rise above 72 beats per minute, 90% of the time.

or in other words

To include individuals in the study, what should the random sample size be so that 90% of the mean samples of this size will be 72 beats per minute or less?

Solution/AnswersExercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

- 1) 2.3%
- 2) 73.29

Solution/AnswersExercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

- 3) 40.96

Other Types of Sampling Distribution

- F distribution

This is a sampling distribution of the mean with an estimated standard deviation.

- t Distribution

This is the sampling distribution of two variances (squared standard deviations).

Application of Sampling distribution

- The sampling distribution like the normal distribution is a descriptive model, so it is used to describe real world situations.
- It is very useful to make statements about the probability of specific observations occurring.
- Investigators/researchers/modelers use it for estimation and hypothesis testing.

References/Further Reading

1) Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

2) Last, J. A Dictionary of Epidemiology. 3rd edition,1995.

3) Wisniewski, M. Quantitative Methods For

Decision Makers, 3rd edition, 2002.

4) Pidd, M. Tools For Thinking. Modelling in Management Science. 2nd edition, 2003.

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