Lesson 10 Meaningful Possibilities Solidify Understanding

Learning Focus

Use simulation to find a plausible interval for the mean of a population from a sample.

How does finding an interval for a population mean compare to finding an interval for a population proportion?

Open Up the Math: Launch, Explore, Discuss

Alyce, Javier, and Veronica are thinking about the original $300$ artifacts they had collected earlier and they are wondering if they can develop a strategy for finding an interval of reasonable values for the average age of the artifacts by taking a sample of $30$ artifacts.

1.

Design and carry out a simulation you could use to create an interval of reasonable values for the actual average age $μ$ of the $300$ artifacts.

2.

Alyce randomly selects $30$ artifacts and finds that $\overset{―}{x} = 999.9$ years and $s x = 13.7699$ years. Using the work from your simulation, design and use a strategy that does not require simulations to find an interval of plausible values for the average age of the artifacts in the bag.

3.

If the average age of the artifacts was really $μ = 990$ years old with $σ = 20$ , how likely would you be to take a sample of $30$ artifacts and get an average age of $1,003.3$ or higher?

4.

Explain why a sample size of $30$ is useful for this problem.

5.

Compare your strategy to the one you developed in the last lesson. Give a general strategy for finding an interval of plausible values for a population parameter using either a sample mean or a sample proportion.

Ready for More?

What could be done to reduce the size of the interval in problem 2 and still have a high degree of confidence that it represents the population mean?

Takeaways

For $n \geq 30,$ $\overset{―}{x} =$ sample mean, $s$ = sample standard deviation

Margin of error for a sample mean:

Interval of plausible values from a sample mean:

Lesson Summary

In this lesson, we found a margin of error for a sample mean, which creates a plausible interval for the population mean. We learned that to use the margin of error formula, we must have a sample size of at least 30. We also learned how to find the likelihood that a sample comes from a population with a given mean and standard deviation.

Retrieval

Given: $A = [\begin{matrix} 1 & 10 & - 1 \\ 2 & 0 & - 2 \end{matrix}] B = [\begin{matrix} - 1 & 4 \\ - 2 & 3 \\ 1 & 0 \end{matrix}]$

1.

Write the dimensions of $A$ and $B$ . Explain how you know whether or not you can multiply these two matrices.

2.

Find $A B$ .

3.

Factor $9 x^{2} - 49$ .

4.

Factor $9 x^{2} - 42 x + 49$ .

5.

Factor $x^{3} - 27$ .