Download - Section 8.3 ~ Estimating Population Proportions Introduction to Probability and Statistics Ms. Young
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Section 8.3 ~ Estimating Population Proportions
Introduction to Probability and StatisticsMs. Young
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Objective
Sec. 8.3
After this section you will learn how to estimate population proportions and compute the associated margins of error and confidence intervals.
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The Basics of Estimating a Population Proportion
The process for estimating a population proportion, p, with a 95% confidence level using a sample proportion, , is the same as the process of estimating a population mean using a sample mean (section 8.2) The only difference is the way that the margin of error is defined:
The confidence interval is written as:
Sec. 8.3
p̂
ˆ ˆ(1 )2E
n
ˆ ˆE E
or
ˆ E
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Example 1
Sec. 8.3
The Nielsen ratings for television use a random sample of households. A Nielsen survey results in an estimate that a women’s World Cup soccer game had 72.3% of the entire viewing audience. Assuming that the sample consists of n = 5,000 randomly selected households, find the margin of error and the 95% confidence interval for this estimate.
The 95% confidence interval is:
0.723 – 0.013 < p < 0.723 + 0.013
or
0.710 < p < 0.736
With 95% confidence, we can conclude that between 71% and 73.6% of the entire viewing audience watched the women’s World Cup soccer game.
ˆ ˆ(1 )2E
n
0.723(1 0.723) 2
5000E
0.013E
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Choosing Sample Size Choosing a sample size appropriate to satisfy a desired margin of
error is found by manipulating this APPROXIMATE formula for margin of error:
Note: any value equal to or larger than the value found using the formula would be sufficient
Sec. 8.3
1 E
n 1E n
1 n
E
21
nE
2
1 n
E
Used to approximate appropriate sample size
2
1 n
E
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Example 2
Sec. 8.3
You plan a survey to estimate the proportion of students on your campus who carry a cell phone regularly. How many students should be in the sample if you want (with 95% confidence) a margin of error of no more than 4 percentage points?
You should survey at least 625 students.
2
1n
E
2
1
0.04n
1 625
.0016n