# 8.6 Practice 1: confidence intervals for averages, known population

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## Student learning outcomes

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

## Given

The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. ( (External Link)

Let $X=$ the age of a Winter Foothill College student

## Calculating the confidence interval

$\overline{x}=$

30.4

$n$ =

25

15=(insert symbol here)

$\sigma$

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

$\overline{X}$ =

the mean age of 25 randomly selected Winter Foothill students

What is $\overline{x}$ estimating?

$\mu$

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

yes

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

Normal

## Explaining the confidence interval

Construct a 95% Confidence Interval for the true mean age of Winter Foothill College students.

How much area is in both tails (combined)? $\alpha =$ $\text{________}$

0.05

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

0.025

Identify the following specifications:

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

• 24.52
• 36.28
• 5.88

The 95% Confidence Interval is:__________________

$\left(\text{24}.\text{52},\text{36}\text{.}\text{28}\right)$

In one complete sentence, explain what the interval means.

## Discussion questions

Using the same mean, standard deviation and level of confidence, suppose that $n$ were 69 instead of 25. Would the error bound become larger or smaller? How do you know?

Using the same mean, standard deviation and sample size, how would the error bound change if the confidence level were reduced to 90%? Why?

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