Overview
8.1 Z Interval for the Mean
8.2 t Interval for the Mean
8.3 Z Interval for a Population Proportion
8.4 Confidence Intervals for the Population
Variance and Standard Deviation
8.5 Sample Size Considerations
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Confidence Intervals for Population Means and Proportions, Lecture notes of Probability and Statistics
An overview of confidence intervals for population means and proportions, including formulas, objectives, and examples. It covers the Z interval for the population mean when σ is known, the t interval for the mean when σ is unknown, and the Z interval for a population proportion. It also discusses the central limit theorem for proportions and the interpretation of confidence intervals.
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2021/2022
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Overview
8.1 Z Interval for the Mean
8.2 t Interval for the Mean
8.3 Z Interval for a Population Proportion
8.4 Confidence Intervals for the Population Variance and Standard Deviation
8.5 Sample Size Considerations
8.1 Z Interval for the Mean
Objectives: By the end of this section, I will be able to…
1)Explain how the formula is developed for the Z interval for the population mean μ. 2)Interpret the meaning of a confidence interval. 3)Calculate and interpret a Z interval for the population mean μ, when σ is known, for two different cases. 4)Explain the meaning of the margin of error.
FIGURE 8.2 Z (^) /2is the value of Z that has area /2 to the right of it for.
The symbol z / 2 represents the value of z in the standard normal distribution so that the area to the right of z is / 2
FIGURE 8.
The symmetry of the bell-curve shows that:
P ( Z / 2 Z Z / 2 ) 1
Solutions
Example
Recall from chapter 7,
Central Limit Theorem
The sampling distribution of the sample mean for a normal population or any population if sample size is at least 30 is distributed as normal with mean:
And standard deviation:
x
x / n
Derive Z Interval for the
Population Mean
Substituting into the left side:
and solving for we get the lower bound for the Z interval for the population mean
z / 2
n
x Z / 2
n
x
Z
/ 2
Derive Z Interval for the
Population Mean
Substituting into the left side:
and solving for we get the upper bound for the Z interval for the population mean
z / 2
n
x Z / 2
n
x
Z
/ 2
Z Interval for the Population
Mean μ
The Z interval can also be written as confidence interval (CI)
n
x Z
n
CI x Z / 2 , / 2
Example
Page 403
Problem 14
z 2 for a 95% Confidence Level
TABLE 8.1 Z /2 values for common confidence levels
Margin of Error
Denoted as E
Measure of the precision of the confidence interval estimate
For the Z interval
n
E Z / 2
Interpreting the Margin of Error
For a (1 - )100% confidence interval for μ
“We can estimate μ to within E units with (1 - )100% confidence.”