Chapter 6: Confidence Intervals
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A confidence interval for a parameter is an interval of numbers within which we expect the true value of the population parameter to be contained. The endpoints of the interval are computed based on sample information.

 

 

 

 

 

 

How confident are we that the true population average is in the shaded area? We are 95% confident. This is the level of confidence.

How many standard errors away from the mean must we go to be 95% confident? From -z to z there is 95% of the normal curve.  

 

There are 4 typical levels of confidence: 99%, 98%, 95% and 90%. Each of the levels of confidence has a different number of standard errors associated with it. We denote this by

 

where a is the total amount of area in the tails of the normal curve. Thus, for a 95% level of confidence

 .

Level of confidence

1-a

a/2
90% 5% 1.645
95% 2.5% 1.96
98% 1% 2.33
99% 0.5% 2.575

How do we compute a confidence interval. After selecting (or being told) that level of confidence, for a large (n>30) sample we use the formula

.

example: A sample of 100 observations is collected and yields m=75 and s=8. Find a 95% confidence interval for the true population average.

 

=

=

=(73.432,76.568).

 

For a sample of 121 observations with an average of 50 and standard deviation of 20, find a 90% confidence interval for the true population average.

 

 

 

Kennesaw State University claims the average starting salary of its graduates is $38,500. A sample of 100 KSU students is sampled and yields an average starting salary of $36,800 with a standard deviation of $9,369. Using a 95% confidence level what can you say about KSU's claim?

 

 

Homework

Section 6-3:  3, 4, 9, 10, 12, 21-23

 

For samples that are not large (i.e. small) we make one correction. We use a t-chart to replace the normal curve chart and use the formula

.

 

example: Find a 99% confidence interval for the true population average from which the sample S={229, 255, 280, 203, 229} is randomly selected.

First we compute the average and standard deviation of the sample. The average is 239.2 and the standard deviation is 29.3. Thus,

=.


degrees of freedom = n-1

Thus, .

==(178.9,299.5)

 

 
Syllabus Files ] Tips for Success ] Intro ] Chapter 2: Numerical Methods ] Chapter 3: Probability ] Chapter 4: Probability Distributions ] Chapter 5: The Normal Curve ] Chapter 5: Central Limit Theorem ] Chapter 4: The Binomial Probability Distribution ] Questionnaire ] Chapter 1 ] [ Chapter 6: Confidence Intervals ] Chapter 6: Required Sample Size ] Chapter 6: Estimation of Proportion ] Required Sample Size (Updated) ] Chapter 7: Hypothesis Testing ] Chapter 7: Applications of Hypothesis Testing ] Sample Tests ]