Section 7.2 Confidence Intervals for the Mean When σ Is Known
Confidence Intervals - Section 7.2
Introduction to Confidence Intervals
Explains the basics of confidence intervals, especially when the population standard deviation (σ) is known.
Covers concepts, calculations, and practical applications of constructing confidence intervals for the mean.
Assumptions for Finding a Confidence Interval
Two assumptions for computing a confidence interval for the mean when σ is known:
The sample must be a random sample.
Either the sample size n ≥ 30 or the population is normally distributed when n < 30.
Robust Statistical Techniques
Some techniques are considered 'robust' and can tolerate slight deviations from normality.
Even with non-perfect normal distribution, confidence intervals can still yield valid conclusions.
Formula for Confidence Interval of Mean (σ Known)
General formula for computing confidence intervals for the mean:
90% confidence interval: 𝑧𝛼/2 = 1.65
95% confidence interval: 𝑧𝛼/2 = 1.96
99% confidence interval: 𝑧𝛼/2 = 2.58
Example 7-1: Skipping Breakfast
Researcher estimates number of times a person skips breakfast annually:
Sample: 49 people
Mean: 50 times
σ = 2.6
95% confidence interval calculation: (49.3, 50.7)
Example 7-2: Amount of Candy Consumed
Nutritionist finds average amount of candy consumed annually:
Sample size: 60 adults
Mean: 23.9 pounds
σ = 4.8
99% confidence interval calculation: (22.3, 25.5)
Diagram for 95% Confidence Interval
Figure 7-3 explains how 95% of the sample means fall within the designated range.
Introduction to Confidence Intervals (continued)
Confidence intervals provide a range for estimating population parameters.
Knowing population standard deviation (σ) allows for higher accuracy in calculations.
Confidence Levels and Critical Values
Confidence levels (90%, 95%, 99%) affect the critical values (z-scores) used in calculations.
Critical value 𝑧𝛼/2 represents the cutoff point for desired confidence levels.
Finding z(α/2) for a 98% Confidence Interval
Set α = 1 - 0.98 = 0.02.
Divide α by 2: α/2 = 0.01 (z-score corresponds to 98% of data in normal distribution).
Using the Standard Normal Table
Locate area closest to 0.9900 (for 98% confidence level).
Find corresponding z-score, approximately 2.33.
When to Use Positive z Scores in Formulas
Only positive z-scores are used in confidence interval formulas due to the symmetry of the normal distribution around the mean.
Normal Distribution and Sample Size
When σ is known and sample size (n) is large (n ≥ 30), standard normal distribution applies.
Central Limit Theorem states distribution of sample means approaches normality as n increases.
Determining Sample Size
Use the formula to determine sample size needed for a specific margin of error (E), ensuring confidence interval meets the desired confidence level.
EXAMPLE 7-3: Credit Union Assets
Data represents random sample of assets (in millions) from 30 credit unions:
Assume population standard deviation is 14.405.
Task: Find the 90% confidence interval of the mean.
Sample data: [12.23, 16.56, 4.39, ...] (30 values total).
SOLUTION Step 1: Mean Calculation
Calculate mean for the data:
Mean X = 11.091
Population standard deviation assumed: 14.405.
SOLUTION Step 2: Finding α/2
For 90% confidence interval: a = 1 - 0.90 = 0.10.
SOLUTION Step 3: Finding Za/2
Z score from normal table for area between 0.9495 and 0.9505 is 1.65 (can also use 1.645 for precision).
SOLUTION Step 4: Substitute in the Formula
Confidence range from:[ 11.091 < 11.091 + 4.339 < 15.430 ]
Interpretation: 90% confident that population mean assets are between $6.752 million and $15.430 million.
Example 7-4: Automobile Thefts
A sociologist estimates average number of automobile thefts per day within a city:
Desired confidence: 99%
Standard deviation: 4.2
Margin of Error: 2
Sample size calculation result suggests to survey 30 days to achieve desired confidence.