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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

• a. x¯=x¯= size 12{ {overline {x}} ={}} {}________________
• b. n=n= size 12{n={}} {}________________
• c. 15 =________ 15=________ (fill in symbol)

#### Solution

• a. 30.4
• b. 25
• c. σxσx size 12{σ rSub { size 8{x} } } {}

### Exercise 2

Define the Random Variable, X¯ X , in words.

X¯ X =_________________

#### Solution

the age of Winter 2006 Foothill students

### Exercise 3

What is x¯x¯ size 12{ {overline {x}} } {} estimating?

μμ size 12{μ} {}

### Exercise 4

Is σxσx size 12{σ rSub { size 8{x} } } {} known?

yes

### Exercise 5

As a result of your answer to (4), state the exact distribution to use when calculating the Confidence Interval.

Normal

## Explaining the Confidence Interval

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

### Exercise 6

How much area is in both tails (combined)? α=α= size 12{α={}} {} ________________

0.05

### Exercise 7

How much area is in each tail? α2=α2= size 12{ { {α} over {2} } ={}} {} ________________

0.025

### Exercise 8

• a. lower limit =__________________
• b. upper limit =__________________
• c. error bound =__________________

• a. 24.52
• b. 36.28
• c. 5.88

### Exercise 9

The 95% Confidence Interval is:__________________

#### Solution

(24,52,36.28)(24,52,36.28) size 12{ $$"24","52","36" "." "28"$$ } {}

### Exercise 11

In one complete sentence, explain what the interval means.

## Discussion Questions

### Exercise 12

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 13

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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