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Confidence Intervals: Practice 3

Module by: Dr. Barbara Illowsky, Susan Dean

Student Learning Outcomes

  • The student will explore the properties of the 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.

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 2

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'={}} {}

Solution 3

  • a. 64
  • b. 80
  • c. 0.08

Exercise 4

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

Solution 4

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?

Solution 5

p p'

Exercise 6

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

Solution 6

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={}} {}

Solution 8

0.80

Exercise 9

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

Solution 9

0.04

Exercise 10

Calculate the following:

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

Solution 10

  • a. 0.72
  • b. 0.88
  • c. 0.08

Exercise 11

The 92% Confidence Interval is:

Solution 11

0.72; 0.88

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 proportion.
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 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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