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
$X$ .
$X$ ~
$B(80,0.80)$
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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