Lesson 9 Sim City Solidify Understanding

Learning Focus

Find an interval that is likely to contain the population proportion from a sample.

How do we know that we found the actual population proportion in a sample distribution?

Open Up the Math: Launch, Explore, Discuss

Alyce, Javier, and Veronica continued to collect additional artifacts around the archeological site. Javier continued to look at the proportion of artifacts that are older than $1,000$ years. As he collected this information, he has noticed that as they look further away from the center tower, the proportion of artifacts older than $1,000$ years decreases. As they were looking for artifacts, Veronica identifies a bag of artifacts that were not identified by the sector they came from. Frustrated, she looks through the bag of artifacts and notices that $17$ of $30$ artifacts in the sample dated older than $1,000$ years.

1.

Which sectors do you believe it is possible the bag came from? Justify your response.

Alyce wonders if they could use some of their work from the other day to help make a prediction about which sector the bag of artifacts came from. She wonders if they could simulate taking samples of size $30$ from each sector they already know and use this to help make a prediction about which sector they came from.

To simulate this, Alyce suggests that they assign people one of the sectors and create bags with chips that represent the proportion of artifacts older than $1,000$ years. For example, you could put $50$ chips in a bag and make $46$ of them black and $4$ of them white. You could then reach your hand in the bag and draw a chip out, replace it, draw another chip out, and repeat this until you had a sample of $30$ artifacts. Just like the work they did the other day, they could take samples of $30$ over and over again and create a distribution of these proportions.

2.

Simulate taking samples of $30$ from the sector your teacher assigns you. Use this data to plot a sampling distribution for your $\hat{p} ’s$ . Conduct the simulation enough times to get a clear picture of the distribution. The tables below provide space for recording your results.

# of artifacts of $1,000$ years old	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
Frequency
Proportion

# of artifacts of $1,000$ years old	$10$	$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$
Frequency
Proportion

# of artifacts of $1,000$ years old	$20$	$21$	$22$	$23$	$24$	$25$	$26$	$27$	$28$	$29$	$30$
Frequency
Proportion

Based on your simulation, do you think it is likely the dropped bag of artifacts came from your sector? Use your results to explain why you think it is or is not plausible that the bag came from your sector.

3.

Looking at the class distributions for the other sectors:

Are there sectors you feel confident the bag did not come from?

Which sectors do you believe the bag could have come from?

How do the graphs of the distributions provide evidence for your claim?

4.

One way of listing the most plausible population proportions for a sample is by using what we call a margin of error. We would list the plausible values as an interval. For example, if you thought it was plausible that the bag of artifacts came from sectors containing proportions of artifacts between $.48$ and $.86$ , you would write this as $(.48, .86)$ , which means it is plausible this sample of artifacts came from any population that had an actual proportion of artifacts more than $1,000$ years old between $.48$ and $.86$ . What do you think would happen to this interval of plausible values for $p$ if the bag contained a larger sample of artifacts? Justify your answer.

5.

Javier is thinking that using simulations to find an interval of plausible values takes time and wonders if their previous work on the Central Limit Theorem and sampling distributions could be useful in finding a way to create an interval of plausible values for the population proportion $p$ . Alyce found another bag of $100$ artifacts that contains $64$ artifacts that date older than $1,000$ years that did not have the sector they were found in recorded.

a.

Design a strategy that you could use to find an interval of reasonable values of $p$ for this bag.

b.

Sector 6

$μ_{\hat{p}} = .36 σ_{\hat{p}} = .047$

Sector 5

$μ_{\hat{p}} = .48 σ_{\hat{p}} = .049$

Sector 4

$μ_{\hat{p}} = .60 σ_{\hat{p}} = .051$

Sector 3

$μ_{\hat{p}} = .74 σ_{\hat{p}} = .044$

Sector 2

$μ_{\hat{p}} = .86 σ_{\hat{p}} = .04$

Sector 1

$μ_{\hat{p}} = .92 σ_{\hat{p}} = .026$

The simulations provided show taking $1, 000$ samples of size $100$ from each sector. Is your interval from above reasonable based on these simulations? Is your interval too wide and include unreasonable values? Is it too narrow and does not include enough reasonable values? Refine your strategy so that it is more likely to give you an interval of reasonable values based on these simulations.

6.

If you were to randomly sample $100$ artifacts from the sector that contained $.92$ artifacts over $1,000$ years old:

a.

How likely are you to get a sample that contains $64$ or fewer artifacts more than $1, 000$ years old?

b.

Based on your answer, is it likely that a bag of $100$ artifacts containing $64$ artifacts older than $1, 000$ years came from this sector? Why or why not?

Ready for More?

What if the sample was only $30$ artifacts from the sector that has $.92$ artifacts more than $1,000$ years old? Can you use the Central Limit Theorem to find the probability that $25$ or fewer artifacts are in this sample? Why or why not?

Takeaways

Margin of error:

Length of the interval:

Vocabulary

interval of plausible values
margin of error
Bold terms are new in this lesson.

Lesson Summary

In the lesson, we learned to find an interval from a sample proportion that is likely to contain the population proportion using a margin of error. To use the formula, we found the sample size must be large enough to satisfy the conditions of the Central Limit Theorem.

Retrieval

1.

Given: $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ . What is special about matrix $A$ ?

2.

Find $\sin (\tan^{- 1} (\sqrt{3}))$ .