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

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

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}} ={}} {}

[ Show Solution ]

Exercise 2

nn=

[ Show Solution ]

Exercise 3

15=(insert symbol here)

[ Show Solution ]

Exercise 4

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

X¯ X =

[ Show Solution ]

Exercise 5

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

[ Show Solution ]

Exercise 6

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

[ Show Solution ]

Exercise 7

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

[ Show Solution ]

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={}} {} ________________

[ Show Solution ]

Exercise 9

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

[ Show Solution ]

Exercise 10

Identify the following specifications:

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

[ Show Solution ]

Exercise 11

The 95% Confidence Interval is:__________________

[ Show Solution ]

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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