# 8.6 Practice 2: confidence intervals for means, 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 $X=$ 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:

• $\overline{x}=$
• ${s}_{x}=$
• $n=$ 

• 3.26
• 1.02
• 39

Define the Random Variable, $\overline{X}$ , in words. $\overline{X}=$ $\text{__________________________}$

the mean number of colors of 39 flags

What is $\overline{x}$ estimating?

$\mu$

Is ${\sigma }_{x}$ known?

No

As a result of your answer to (4), state the exact distribution to use when calculating the Confidence Interval.

${t}_{\text{38}}$

## 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)? $\alpha =$

0.05

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

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.

In one complete sentence, explain what the interval means.

## Discussion questions

Using the same $\overline{x}$ , ${s}_{x}$ , and level of confidence, suppose that $n$ were 69 instead of 39. Would the error bound become larger or smaller? How do you know?

Using the same $\overline{x}$ , ${s}_{x}$ , and $n=\text{39}$ , how would the error bound change if the confidence level were reduced to 90%? Why?

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