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Textbook by: Kathy Chu, Ph.D.. E-mail the author

# Practice 1: Confidence Intervals for Averages, Known Population Standard Deviation

## Student Learning Outcomes

• The student will calculate confidence intervals for means when the population standard deviation is known.

## Given

The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. (http://research.fhda.edu/factbook/FH_Demo_Trends/FoothillDemographicTrends.htm

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

## Calculating the Confidence Interval

### Exercise 1

x¯=x¯= size 12{ {overline {x}} ={}} {}

30.4

nn=

25

### Exercise 3

15=(insert symbol here)

σσ

### Exercise 4

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

X¯ X =

#### Solution

the mean age of 25 randomly selected Winter Foothill students

### Exercise 5

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

μμ size 12{μ} {}

### Exercise 6

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

yes

### Exercise 7

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 mean age of Winter Foothill College students.

### Exercise 8

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

0.05

### Exercise 9

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

0.025

### Exercise 10

Identify the following specifications:

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

• a. 24.52
• b. 36.28
• c. 5.88

### Exercise 11

The 95% Confidence Interval is:__________________

#### Solution

(24.52,36.28)(24.52,36.28) size 12{ $$"24","52","36" "." "28"$$ } {}

### Exercise 13

In one complete sentence, explain what the interval means.

## Discussion Questions

### Exercise 14

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 15

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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