Sampling Distributions

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Sampling Distributions. Introduction. Discrete distributions Binomial Poisson Hypergeometric Continuous distributions Normal We knew µ and used it to determine P(X) Using sample data to project to the population – inferential statistics. Preview.

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### Sampling Distributions

Introduction
• Discrete distributions
• Binomial
• Poisson
• Hypergeometric
• Continuous distributions
• Normal
• We knew µ and used it to determine P(X)
• Using sample data to project to the population – inferential statistics
Preview
• Consider the sample mean as a random variable
• Different values for sample mean
• Own mean and s.d.
• Probability distribution of sample means
• Sample distribution of the mean
The Population

µ = (1+1+3+5)/4 = 2.5

Sampling Distribution of the Mean

= (1.0) + (2.0) + (3.0) + (4.0) = 2.5

Sampling Distribution of the Mean

The sampling distribution of the mean will always have the same mean as the original population.

The standard deviation of the sampling distribution of the mean is referred to as the standard error of the mean.

Sampling Distribution of the Mean

If the original population is distributed normally, the sampling population will also be normal.

If the original population is not normal, the sampling distribution will approximate normal.

Sampling Distribution of the Proportion

Sometimes, the results of our sampling are best expressed as a proportion:

Population Normally Distributed
• Regardless of sample size, sampling distribution of the mean will be normally distributed
• Mean:
• Standard error of the mean:
• Where:
• μ = population mean
• σ = population standard deviation
• n = sample size
Sampling Distribution of the Mean

Combining the results we just got, describe the sampling distribution of the mean.

Determine the mean and standard deviation of x.

For the sample size n=2, determine the mean for each simple random sample from the population.

For each sample we just identified, what is the probability this sample will be selected?

Do this again for n=3. What effect does the change in sample size have on the mean and the standard error of the mean?

Given the following probability distribution for an infinite population with the discrete RV, x:

Effect of Sample Size

The average annual hours flown by general aviation aircraft = 130.

Assume these hours are normally distributed.

Standard deviation = 30 hours.

Practice

A random variable is normally distributed with μ = \$1500 and σ = \$100. Determine the standard error of the mean for simple random samples with the following sample sizes:

n=16

n=100

n=400

n=1000

Conversion to Standard Normal
• z-score for sampling distribution of the mean:
• Where:
• z = distance from the mean measure in standard error units
• = value of the sample mean we’re looking for
• μ = population mean
• = standard error of the mean
Z-score Example

Airplanes with μ = 130 and σ = 30

For a simple random sample of 36 aircraft, what is the probability the average flight time for the aircraft in the sample was ≥ 138 hours?

Picking up where we left off…

A crane is operated by 4 electrical motors working together. For the crane to work properly, the 4 motors must generate 380 hp.

Each motor produces an average of 100 hp with a standard deviation of 10 hp.

What’s the probability that the crane won’t work?

Central Limit Theorem
• CLT for the sampling distribution of the mean:
• = µ
• When n is large, the sampling distribution of will be approximately normal.
• When the population distribution is normal, the sampling distribution will be normal for any sample size.
Practice

A store receives a truckload of electric components. Before accepting shipment, the store will randomly select 9 components for testing.

The shipment will be rejected if the components resistance is > 300Ω as listed on their label.

The true mean of the population is 295Ω, the σ is 12Ω, and the population is normally distributed.

What is the probability the load will be rejected?

Practice

The average length of a hospital stay is 5.7 days. Assuming a σ of 2.5 days and a random sample of 50 patients, what is the probability the average stay for the sample will be ≤ 6.5 days?

If the sample had been 8 instead of 50, what further assumption must we make in order to solve this problem?

Example

What is the probability that a single, random reading will show this patient with BP above 150?

What is the probability that the sample mean from 5 readings will show this patient with BP above 150?

How many samples must we take before the probability of the sample mean being less than 150 is .01?

A patient with a systolic blood pressure (BP) above 150 is considered to have high BP and may require medical treatment.

Any patients BP can vary greatly during a normal day.

Suppose a patient’s BP has µ = 160, = 20

Proportions
• Sometimes the variable we’re interested in is a proportion, e.g.:
• Proportion of defectives in a lot
• Proportion of incoming freshman who graduate within 5 years.
• Symbols
• p = population proportion
• = sample proportion
Proportions

Generally, the numerator is the number of interest, the denominator is the number in the sample:

Like sample mean, sample proportion estimates the population proportion

Properties of the Sample Proportion

Sampling distribution of the proportion:

When nπand n(1- π)are ≥ 5, the normal distribution can be applied

Properties of the Sample Proportion

Standard error of the proportion:

z-score for the proportion

Example

According to the Bureau of the Census, 85.1% of Arizona’s adult residents have completed high school.

What is the probability that ≤ 80% of a random sample of 200 Arizona adults have finished high school (the probability that ≤ .80?

Practice

What is the population proportion?

What is the sample proportion?

What is the standard error of the sample proportion?

If we took another random sample, what’s the probability we’d get a proportion at least as large (.35) as the one we got here?

42.6% of all purchasing agents in the U.S. work force are women. In a random sample of 200 purchasing agents, 70 are women.

And finally…Finite Populations

When sampling w/out replacement from a finite population, the sampling error (for mean or proportion) becomes small when n is 5% or more of N.

For the mean:

For the proportion:

Example

Of the 629 imported cars sold in Enterprise last year, 117 were Toyotas.

A random sample of 300 imported cars is conducted.

What is the probability that at least 15% of the vehicles in the sample will be Toyotas?