Lesson 8 Ancient Treasures Solidify Understanding

Learning Focus

Use random sampling methods and find means and proportions for random samples.

How do researchers find averages and percentages for a population when they can’t measure the entire population?

Open Up the Math: Launch, Explore, Discuss

Alyce, Javier, and Veronica have made a significant amount of progress at the archeological site after staking it out. They have been slowly uncovering artifacts in the different sectors and have a collection of $300$ artifacts that have been dated.

A list of the artifacts and their age is given in a table, and a histogram of the data is shown here:

1.

What observations do you have about the artifacts based on the data set and the histogram? What questions might you ask about the data?

As Alyce is looking at the data, she is wondering about two questions: What is the average age of the artifacts, and what proportion of these artifacts are over $1,000$ years old? There is a lot of data and she really does not want to find the average or proportion by using all $300$ artifacts. She decides that she could estimate the average age $μ$ and the proportion $p$ by taking a sample of the $300$ artifacts. Javier thinks this is a good idea but also suggests that maybe they could get a better estimate if they each took a sample and found the average of those artifacts, and the proportion of those artifacts that are over $1,000$ years old and compared them to each other. He suggests they each take a sample of $5$ and find the average.

2.

a.

Select and describe a sampling method you could use to select a random sample of $5$ artifacts from the list.

b.

Using your sampling method, randomly select a sample of $5$ artifacts and write down the average and the proportion of your sample.

${\overset{―}{x}}_{5} (the sample average) =$

${\hat{p}}_{5} (the sample proportion) =$

c.

Based on your sample, make a prediction for the actual average age of all the artifacts.

3.

To help the archeologists, take $3$ samples of $5$ artifacts and record their ages. Find the average age and the proportion of artifacts older than $1,000$ years in each sample and record them in the table below. As you finish finding your sample means and proportions, add them to the class graph.

Average Age

${\overset{―}{x}}_{5}$

Proportion of Artifacts Older than $1,000$ Years in Sample

${\hat{p}}_{5}$

Sample 1

Sample 2

Sample 3

As Alyce looks at the graph created, she notices that it looks a little different from the graph of the original $300$ artifacts.

4.

Compare this new graph with the graph for the population of all $300$ artifacts. What is the same? What changed?

5.

Explain the difference between how this new graph was created and how the graph for the $300$ artifacts was created.

Pause and Reflect

As Veronica is looking at the graphs, she is still wondering about using only $5$ artifacts to make a prediction about the average for all $300$ . She says, “I am curious how these graphs would change if we did this for a larger sample than $5$ . What if we sampled $10$ instead? Or what if we sampled $30$ ?” Javier suggests they repeat what they just did for larger sample sizes and compare those graphs too.

6.

Take $3$ samples of size $10$ and then $3$ samples of size $30$ . Share your results with the rest of the class and create graphs for samples of size $10$ and $30$ .

Average Age

${\overset{―}{x}}_{10}$

Proportion of Artifacts Older than $1,000$ Years in Sample

${\hat{p}}_{10}$

Sample 1

Sample 2

Sample 3

Average Age

${\overset{―}{x}}_{30}$

Proportion of Artifacts Older than $1, 000$ Years in Sample

${\hat{p}}_{30}$

Sample 1

Sample 2

Sample 3

7.

Compare and contrast the distributions of these graphs with the original and write your observations.

Pause and Reflect

8.

If you were to take one sample and use that one sample to predict the average age of all $300$ artifacts, would taking a sample of $5$ , $10$ , or $30$ be better? Use the graphs to justify your answer.

Ready for More?

Give an example where it might be nearly impossible to find the population mean, but it would be possible to take random samples to find a sample mean.

Takeaways

Central Limit Theorem:

Given a population with mean $μ$ and standard deviation $σ$ :

Adding Notation, Vocabulary, and Conventions

Parameter:

Population mean:

Population proportion:

Sample statistic:

Sample mean:

Sample proportion:

Vocabulary

Lesson Summary

In this lesson, we sampled a population and used the mean and proportion of the population to estimate the actual mean and proportion of the sample, which are called population parameters. We learned about the Central Limit Theorem, which says that if the sample is large enough, the sample means and proportions will be normally distributed.

Retrieval

1.

Let $A = [\begin{matrix} - 7 & - 4 \\ - 3 & 5 \end{matrix}]$ and $B = [\begin{matrix} 7 & 4 \\ 3 & - 5 \end{matrix}]$

What is the name of the matrix that is the sum of A + B?

2.

Given: $θ = \tan^{- 1} (- 2.145)$ . Find two angles $(0 < θ \leq 2 π)$ that make the equation true.