# 8.5 Practice 1: confidence intervals for means, 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?

#### Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
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ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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