# 9.7 Practice 3: confidence intervals for proportions

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## 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

What is being counted?

In words, define the Random Variable $X$ . $X=$

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

Calculate the following:

• $x=$
• $n=$
• $p\text{'}=$

• 64
• 80
• 0.8

State the estimated distribution of $X$ . $X$ ~

$B\left(80,0.80\right)$

Define a new Random Variable $P\text{'}$ . What is $p\text{'}$ estimating?

$p$

In words, define the Random Variable $P\text{'}$ . $P\text{'}=$

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

State the estimated distribution of $P\text{'}$ . $P\text{'}$ ~

## 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.

How much area is in both tails (combined)? $\alpha =$

1 - 0.92 = 0.08

How much area is in each tail? $\frac{\alpha }{2}=$

0.04

Calculate the following:

• lower limit =
• upper limit =
• error bound =

• 0.72
• 0.88
• 0.08

The 92% Confidence Interval is:

(0.72; 0.88)

In one complete sentence, explain what the interval means.

## Discussion questions

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

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

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

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