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Practice 2: Confidence Intervals for Averages, Unknown 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 unknown.

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 mean number of colors on a national flag. Let X=X= size 12{X={}} {} the number of colors on a national flag.

Table 1
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

  • 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

the mean number of colors of 39 flags

Exercise 3

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

Solution

μ μ size 12{μ} {}

Exercise 4

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

Solution

No

Exercise 5

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

Solution

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

Confidence Interval for the True Mean Number

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

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

Calculate the following:

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

Solution

  • a. 2.93
  • b. 3.59
  • c. 0.33

Exercise 9

The 95% Confidence Interval is:

Solution

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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