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8.6 Practice 2: confidence intervals for averages, unknown population

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Student learning outcomes

  • The student will calculate confidence intervals for means when the population standard deviation is unknown.

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 mean 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:

  • x ¯ = size 12{ {overline {x}} ={}} {}
  • s x = size 12{s rSub { size 8{x} } ={}} {}
  • n = size 12{n={}} {}

  • 3.26
  • 1.02
  • 39

Define the Random Variable, X ¯ size 12{ {overline {X}} } {} , in words. X ¯ = size 12{ {overline {X}} ={}} {} __________________________

the mean number of colors of 39 flags

What is x ¯ size 12{ {overline {x}} } {} estimating?

μ size 12{μ} {}

Is σ x 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 mean number

Construct a 95% Confidence Interval for the true mean 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? α 2 = size 12{ { {α} over {2} } ={}} {}

0.025

Calculate the following:

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

  • 2.93
  • 3.59
  • 0.33

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.

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.

In one complete sentence, explain what the interval means.

Discussion questions

Using the same x ¯ size 12{ {overline {x}} } {} , s x 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}} } {} , s x 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?
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Source:  OpenStax, Collaborative statistics: custom version modified by r. bloom. OpenStax CNX. Nov 15, 2010 Download for free at http://legacy.cnx.org/content/col10617/1.4
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