COMMERCE 2QA3 Study Guide - Final Guide: Confidence Interval, Simple Random Sample, Sampling Distribution
28 Jun 2018
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Modelling Sample Proportions
True proportion is denoted as as a value of an event in a population. While we can only see the sample
actually drawn, it is possible to imagine the results of all other possible samples. A simulation uses computers to
pretend to draw random samples from a population, helping us to understand how sample proportions vary
depending on the random sampling.
When there are only two possible outcomes, one is the "success" and the other "failure." In a simulation, the
true proportion of successes is set and random samples are drawn, where the sample proportion of successes is
recorded. This is denoted by
for each sample.
Sampling Distributions for Proportions
The distribution of proportions over many independent samples from the same population is the sampling
distribution. For bell-shaped distributions centred at true proportion, the sample size n is used to find the
standard deviation:
Where n is the sample size and q is the proportion of failure
is the observed value). The particular Normal
mode,
is the sampling distribution model for the sample proportion.
The difference between sample proportions, the sampling error is not truly an error but a variability - sampling
variability. There are several assumptions and conditions -
Independence
Sampled values must be independent
Sample size
Sample size n must be large enough
Randomization
Subjects should have been randomly assigned - simple random sample
10% condition
If sampling not made with replacement, sample size n must be smaller than 10% of the
population
Success/failure
condition
Sample size must be big enough that np (number of successes) and nq (number of
failures) are expected to be at least 10
Central Limit Theorem
The sampling distribution (sample means and proportions) of any mean becomes NORMAL as the sample size
grows, regardless of the population distribution. The larger the sample size, the more closely the Normal
approximates the sampling distribution model for the mean.
Sampling Distribution of the Mean
Means have smaller standard deviations than individuals. When a random sample is drawn from any population
with mean and standard deviation , its sample mean
has the same mean but a different standard deviation
of
.
Standard Error
This is the estimated standard deviation of a sampling distribution.
For a sample proportion
The standard error is
Margin of Error: Certainty and Precision
Our confidence interval can be expressed as below as
.
The margin of error is the extent of the interval on either side of
.
The general condfidence interval is expressed in terms of ME:
Sampling Distributions
September 27, 2017
4:36 PM
Statistics Page 1
Document Summary
True proportion is denoted as as a value of an event in a population. While we can only see the sample actually drawn, it is possible to imagine the results of all other possible samples. A simulation uses computers to pretend to draw random samples from a population, helping us to understand how sample proportions vary depending on the random sampling. When there are only two possible outcomes, one is the "success" and the other "failure. " In a simulation, the true proportion of successes is set and random samples are drawn, where the sample proportion of successes is. The distribution of proportions over many independent samples from the same population is the sampling distribution. For bell-shaped distributions centred at true proportion, the sample size n is used to find the standard deviation: recorded. Where n is the sample size and q is the proportion of failure is the observed value).
