Polls and Margin of Error
Section 4.5
Public opinion polls are used to gather information about a population’s views, expectations, likes and dislikes.
Sampling and Inferential Statistics.
The larger the sample, the more likely it will reflect the population. However, costs and feasibility are major factors in determining size of a sample. If 100 people polled and 79 responded yes to the question, “Is there too much violence on TV?” how confident is a pollster with his or her results? Do the sample data reflect the general population? Does that mean that 79% of the entire population think there is too much violence on TV? By gathering data from different samples, another 100 might answer 72 yes or 68 yes to violence on TV.
The set of all sample proportions and probabilities can be represented by a bell-shaped curve. A sample estimate is not 100% accurate, although it comes close to the true population, there is an error associated with the sample. Using the bell curve, we can predict the probable error of a sample estimate.
______________________|_______________________ x/n
True Population Sample
Proportion proportion
In order to predict error, there is new terminology used
In a standard normal distribution, we use a z-distribution body table.
Alpha = a = area from P(0 < z < z a )
z a is the value of z that yields area a .
P(0 < z < z a ) = a
Using a differentiates sample proportion from other normal distributions.
Example 1: Find
a) z0.3810 means find P(0 < z < z0.3810 ) = 1.18 on the body table
b) z0.47 The body table does not have 0.47 exactly, but does have 0.4699, so we use this value of z = 1.88
c) z0.4954 .4954 comes between .4953 and .4955 so half-way between z = 2.60 and z = 2.61 will yield z = 2.605
Sample proportions x/n vary from sample to sample; some will have small errors and some large errors. Knowing that errors are inherent with sample estimates, the margin of error (MOE) is a prediction of errors with sample estimates.
MOE = z
a
/2
The reason we use
a/2
is that the body table uses .5 data.
2
MOE depends on two factors, sample size, n, and level of confidence a .
Example 2:
a) Find MOE when 500 people are polled and level of confidence is 90%.
N = 500, a = .90 , a /2 = .45 z .45 = 1.645
MOE = 1.645 = .0372 ~ 3.7% Margin of Error
2
b) Find MOE when 1000 people are polled and level of confidence is 90%.
N = 1000, a = .90, a /2 = .45 z .45 = 1.645
MOE = 1.645 =
.0260 ~ 2.6%
Margin of Error
2
Notice the larger n is (sample size), the smaller the MOE.
Example 3:
In a survey of 500 undergraduate students, 410 responded that they will graduate in 4 years.
a) Find the sample proportion that responded to graduating in 4 years.
410/500 = 82%
b) At 95% confidence level, find MOE.
a = .95 a /2 = .475 z.475 = 1.96 n = 500
MOE = 1.96
= .0438 ~
4.4%
2
So we are 95% confident that 82%
4.4% believe they will graduate in 4 years.
From 77.6% to 86.4% believe they will graduate in 4 years.
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