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Confidence Intervals: Practice 1

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

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.

Note: You are viewing an old version of this document. The latest version is available here.

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?

Solution

μμ size 12{μ} {}

Exercise 4

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

Solution

yes

Exercise 5

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 6

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

Solution

0.05

Exercise 7

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

Solution

0.025

Exercise 8

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

Solution

  • 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 10

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 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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