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Confidence Intervals: Practice 1

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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Student Learning Outcomes

  • The student will explore the properties of Confidence Intervals for Averages when the population standard deviation is known.

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

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

Solution

30.4

Exercise 2

nn=

Solution

25

Exercise 3

15=(insert symbol here)

Solution

σσ

Exercise 4

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

X¯ X =

Solution

the age of Winter 2006 Foothill students

Exercise 5

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

Solution

μμ size 12{μ} {}

Exercise 6

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

Solution

yes

Exercise 7

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

Solution

Normal

Explaining the Confidence Interval

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

Exercise 8

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

Solution

0.05

Exercise 9

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

Solution

0.025

Exercise 10

Identify the following specifications:

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

Solution

  • 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 12

Figure 1
Fill in the blanks on the graph with the areas, upper and lower limits of the Confidence Interval, and the sample mean.
Normal distribution curve with two vertical upward lines from the x-axis to the curve. The confidence interval is between these two lines. The residual areas are on either side.

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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