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# Practice 3: Confidence Intervals for Proportions

## Student Learning Outcomes

• The student will calculate confidence intervals for proportions.

## Given

The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 - 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 - 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class is a random sample of the population.

## Estimated Distribution

### Exercise 1

What is being counted?

### Exercise 2

In words, define the Random Variable XX size 12{X} {}. X=X= size 12{X={}} {}

#### Solution

The number of girls, age 8-12, in the beginning ice skating class

### Exercise 3

Calculate the following:

• a. x=x= size 12{x={}} {}
• b. n=n= size 12{n={}} {}
• c. p'=p'= size 12{p'={}} {}

• a. 64
• b. 80
• c. 0.8

### Exercise 4

State the estimated distribution of XX size 12{X} {}. XX ~

#### Solution

B ( 80 , 0.80 ) B(80,0.80)

### Exercise 5

Define a new Random Variable P'P' size 12{P'} {}. What is p'p' size 12{p'} {} estimating?

p p

### Exercise 6

In words, define the Random Variable P'P' size 12{P'} {} . P'=P'= size 12{P'={}} {}

#### Solution

The proportion of girls, age 8-12, in the beginning ice skating class.

### Exercise 7

State the estimated distribution of P'P' size 12{P'} {}. P'P' ~

## Explaining the Confidence Interval

Construct a 92% Confidence Interval for the true proportion of girls in the age 8 - 12 beginning ice-skating classes at the Ice Chalet.

### Exercise 8

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

1 - 0.92 = 0.08

### Exercise 9

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

0.04

### Exercise 10

Calculate the following:

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

• a. 0.72
• b. 0.88
• c. 0.08

### Exercise 11

The 92% Confidence Interval is:

(0.72; 0.88)

### Exercise 13

In one complete sentence, explain what the interval means.

## Discussion Questions

### Exercise 14

Using the same p'p' size 12{p'} {} and level of confidence, suppose that n were increased to 100. Would the error bound become larger or smaller? How do you know?

### Exercise 15

Using the same p'p' size 12{p'} {} and n=80n=80 size 12{n="80"} {}, how would the error bound change if the confidence level were increased to 98%? Why?

### Exercise 16

If you decreased the allowable error bound, why would the minimum sample size increase (keeping the same level of confidence)?

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