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Confidence Intervals: Practice 2

Module by: Susan Dean, Dr. Barbara Illowsky

Student Learning Outcomes

  • The student will explore the properties of confidence intervals for averages, as well as the properties of an unknown population standard deviation.

Given

The following real data are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true average number of colors on a national flag. Let X=X= size 12{X={}} {} the number of colors on a national flag.

X Freq.
1 1
2 7
3 18
4 7
5 6

Calculating the Confidence Interval

Exercise 1

Calculate the following:

  • a. x¯=x¯= size 12{ {overline {x}} ={}} {}
  • b. sx=sx= size 12{s rSub { size 8{x} } ={}} {}
  • c. n=n= size 12{n={}} {}

Solution 1

  • a. 3.26
  • b. 1.02
  • c. 39

Exercise 2

Define the Random Variable, X¯X¯ size 12{ {overline {X}} } {}, in words. X¯=X¯= size 12{ {overline {X}} ={}} {} ____________________________________________________

Solution 2

the average number of colors of 39 flags

Exercise 3

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

Solution 3

μ μ size 12{μ} {}

Exercise 4

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

Solution 4

No

Exercise 5

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

Solution 5

t 38 t 38 size 12{t rSub { size 8{"38"} } } {}

Confidence Interval for the True Average Number

Construct a 95% Confidence Interval for the true average number of colors on national flags.

Exercise 6

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

Solution 6

0.05

Exercise 7

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

Solution 7

0.025

Exercise 8

Calculate the following:

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

Solution 8

  • a. 2.93
  • b. 3.59
  • c. 0.33

Exercise 9

The 95% Confidence Interval is:

Solution 9

2.93; 3.59

Exercise 10

Fill in the blanks on the graph with the areas, upper and lower limits of the Confidence Interval, and the sample mean.

Figure 1
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 x¯x¯ size 12{ {overline {x}} } {}, sxsx size 12{s rSub { size 8{x} } } {}, and level of confidence, suppose that nn size 12{n} {} were 69 instead of 39. Would the error bound become larger or smaller? How do you know?

Exercise 13

Using the same x¯x¯ size 12{ {overline {x}} } {}, sxsx size 12{s rSub { size 8{x} } } {}, and n=39n=39 size 12{n="39"} {}, how would the error bound change if the confidence level were reduced to 90%? Why?

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