Student Learning Outcomes The student will explore the properties of confidence intervals for averages, as well as the properties of an unknown population standard deviation.

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 average number of colors on a national flag. Let X= size 12{X={}} {} the number of colors on a national flag. X Freq. 1 1 2 7 3 18 4 7 5 6

Calculating the Confidence Interval Calculate the following: a x¯= size 12{ {overline {x}} ={}} {} bsx= size 12{s rSub { size 8{x} } ={}} {} cn= size 12{n={}} {} a3.26 b1.02 c39 Define the Random Variable, X¯ size 12{ {overline {X}} } {}, in words. X¯= size 12{ {overline {X}} ={}} {} __________________________ the average number of colors of 39 flags What is x¯ size 12{ {overline {x}} } {} estimating? μ size 12{μ} {} Is σx size 12{σ rSub { size 8{x} } } {} known? No As a result of your answer to (4), state the exact distribution to use when calculating the Confidence Interval. t 38 size 12{t rSub { size 8{"38"} } } {}

Confidence Interval for the True Average Number Construct a 95% Confidence Interval for the true average number of colors on national flags. How much area is in both tails (combined)? α= size 12{α={}} {} 0.05 How much area is in each tail? α2= size 12{ { {α} over {2} } ={}} {} 0.025 Calculate the following: alower limit = bupper limit = cerror bound = a 2.93 b3.59 c0.33 The 95% Confidence Interval is: 2.93; 3.59 Fill in the blanks on the graph with the areas, upper and lower limits of the Confidence Interval, and the sample mean.

In one complete sentence, explain what the interval means.

Discussion Questions Using the same x¯ size 12{ {overline {x}} } {}, sx size 12{s rSub { size 8{x} } } {}, and level of confidence, suppose that n size 12{n} {} were 69 instead of 39. Would the error bound become larger or smaller? How do you know? Using the same x¯ size 12{ {overline {x}} } {}, sx size 12{s rSub { size 8{x} } } {}, and n=39 size 12{n="39"} {}, how would the error bound change if the confidence level were reduced to 90%? Why?