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# Practice 2: Confidence Intervals for Averages, Unknown Population Standard Deviation

## 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={}} {}

• 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?

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={}} {}

0.05

### Exercise 7

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

0.025

### Exercise 8

Calculate the following:

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

• a. 2.93
• b. 3.59
• c. 0.33

### Exercise 9

The 95% Confidence Interval is:

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.

### 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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