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# Confidence Intervals: Practice 1

Summary: Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

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## Student Learning Outcomes

The student will explore the properties of Confidence Intervals for Averages and gain knowledge of population standard deviation.

## Given

The average age for all Foothill College students for Fall 2005 was 32.7. The population standard deviation has been pretty consistent at 15. Twenty-five Winter 2006 students were randomly selected. The average age for the sample was 30.4. We are interested in the true average age for Winter 2006 Foothill College students. (http://research.fhda.edu/factbook/FHdemofs/Fact_sheet_fh_2005f.pdf)

Let X=X= size 12{X={}} {} the age of a Winter 2006 Foothill College student

## Calculating the Confidence Interval

### Exercise 1

1. x¯=x¯= size 12{ {overline {x}} ={}} {}__________ n=n= size 12{n={}} {} ________ 15 = ___________ (fill in symbol)
2. Define the Random Variable, X¯X¯ size 12{ {overline {X}} } {} , in words. X¯=X¯= size 12{ {overline {X}} ={}} {} ___________________________________________________________
3. What is x¯x¯ size 12{ {overline {x}} } {} estimating?
4. Is σxσx size 12{σ rSub { size 8{x} } } {} known?
5. As a result of your answer to (4), state the exact distribution to use when calculating the Confidence Interval.

#### Solution

1. 30.4; 25; σxσx size 12{σ rSub { size 8{x} } } {}
2. the age of Winter 2006 Foothill students
3. μμ size 12{μ} {}
4. yes
5. Normal

## Explaining the Confidence Interval

### Exercise 2

Construct a 95% Confidence Interval for the true average age of Winter 2006 Foothill College students.

1. How much area is in both tails (combined)? α=α= size 12{α={}} {} __________
2. How much area is in each tail? α2=α2= size 12{ { {α} over {2} } ={}} {}__________
3. lower limit = ________ upper limit = ________ error bound = ________
4. The 95% Confidence Interval is: _____________________
5. Fill in the blanks on the graph with the areas, upper and lower limits of the Confidence Interval, and the sample mean.
6. In one complete sentence, explain what the interval means.

#### Solution

1. 0.05
2. 0.025
3. 24.52; 36.28; 5.88
4. (24,52,36.28)(24,52,36.28) size 12{ $$"24","52","36" "." "28"$$ } {}

## Discussion Questions

### Exercise 3

Using the same mean, standard deviation and level of confidence, suppose that nn size 12{n} {} were 69 instead of 25. Would the error bound become larger or smaller? How do you know?

### Exercise 4

Using the same mean, standard deviation and sample size, how would the error bound change if the confidence level were reduced to 90%? Why?

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