Preparing for the next class
In the next in-class activity, you will need to understand how sample size affects margin of error and be able to determine the sample size needed to achieve a given margin of error when working with proportions.
In this preview assignment, you will be building on concepts you learned in In-Class Activities 10.A and 10.B. Refer to those activities if you need to review any vocabulary or skills.
Go to https://dcmathpathways.shinyapps.io/SampDist_prop/ and open the DCMP Sampling Distribution of the Sample Proportion tool. Check the box for the option to enter numerical values for [latex]n[/latex] and [latex]p[/latex] and show summary statistics.
Question 1
Set the population proportion to 0.3. For each sample size in the table below, draw 100 samples. Note the mean [latex]\hat{p}[/latex] and approximate minimum and maximum [latex]\hat{p}[/latex] generated for each sample size.
| Sample Size | Mean [latex]\hat{p}[/latex]% | Minimum [latex]\hat{p}[/latex]% | Maximum [latex]\hat{p}[/latex]% |
| 50 | |||
| 100 | |||
| 500 |
Hint: Use the summary statistics found at the bottom of the page. Use the “Select Range of x-axis (zoom in)” feature to zoom in as needed.
Question 2
Which of the following statements best describes the changes you observed as the sample size increased?
- The mean [latex]\hat{p}[/latex]
- The mean [latex]\hat{p}[/latex]
- The interval between the minimum and maximum [latex]\hat{p}[/latex]
- The interval between the minimum and maximum [latex]\hat{p}[/latex]
Hint: Look at the difference between the minimum and maximum [latex]\hat{p}[/latex]. Create additional tables if you need to see the changes again.
Recall that the equation for margin of error ([latex]E[/latex]) is [latex]E = z^{*} \bullet (standard~error)[/latex] and the equation for standard error for a proportion is [latex]\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/latex], where [latex]\hat{p}[/latex] is the sample proportion and [latex]n[/latex] is the sample size.
Question 3
Look at the formula for margin of error. Which of the following statements best describes how you expect the margin of error ([latex]E[/latex]) to change as the sample size ([latex]n[/latex]) increases?
- The margin of error will increase.
- The margin of error will decrease.
- The margin of error will stay the same.
Hint: What happens to the value of a fraction as the denominator increases?
Question 4
Now, let’s test your hypothesis.
- Go to https://dcmathpathways.shinyapps.io/Inference_prop/ to open the Inference for a Proportion tool. Change the “Enter Data” setting to “Number of Successes.” Using a 95% confidence interval, use the tool to complete the table below.
Sample Size # of Successes Point Estimate Margin of Error 100 30 1,000 300 10,000 3,000 Hint: Enter the sample size and number of successes into the web tool.
- Based on your findings, determine whether this statement is true or false:
As long as the point estimate and confidence level remain the same, the absolute value of margin of error will decrease as the sample size increases.
It is important to note that the formula [latex]E = z^{*} \bullet (standard~error)[/latex] is valid only if certain conditions are met:
- Random sampling is used.
- The sample is less than 10% of the population.
• The sample is large enough that [latex]n\hat{p} \geq 10[/latex] and [latex]n(1-\hat{p}) \geq 10[/latex].
Question 5
For each of the following combinations of [latex]n[/latex] and [latex]\hat{p}[/latex], determine whether it would be appropriate to use the formula for the margin of error, [latex]E[/latex]. Assume each sample is random and less than 10% of the population.
| [latex]n[/latex] | [latex]\hat{p}[/latex]% | Appropriate?
(Yes or no) |
| 50 | 0.1 | |
| 100 | 0.1 | |
| 45 | 0.5 | |
| 30 | 0.5 | |
| 25 | 0.9 | |
| 120 | 0.9 |
Hint: Are [latex]n\hat{p} \geq 10[/latex] and [latex]n(1-\hat{p} \geq 10[/latex]?
Researchers can use the same formula to determine the minimum sample size needed to produce a given margin of error simply by solving for [latex]n[/latex]. The rearranged formula looks like this:
[latex]n = \hat{p}(1-\hat{p})(\frac{z^{*}}{E})^{2}[/latex]
Notice that this formula requires the researcher to know the value of [latex]\hat{p}[/latex], which is unknown. However, researchers often have preliminary data or prior research that can be used to estimate [latex]\hat{p}[/latex].
If there is no way to estimate [latex]\hat{p}[/latex], researchers will find the largest possible n by setting [latex]\hat{p}[/latex] to 0.5. (Try it! The largest value you can get for [latex]\hat{p}(1-\hat{p})[/latex]) is 0.25 when you set [latex]\hat{p}[/latex] to 0.5.)
Question 6
Go to https://dcmathpathways.shinyapps.io/Inference_prop/ to open the DCMP Inference for a Proportion tool. At the top of the page, click “Find Sample Size.” Select “Use Conservative Approach [latex]\hat{p}[/latex].”
What sample sizes would be needed for the following confidence level and margin of error combinations?
| Confidence Level | Margin of Error | Minimum Sample Size |
| 95% | [latex]\pm[/latex]2.5% | |
| 95% | [latex]\pm[/latex]4% | |
| 95% | [latex]\pm[/latex]6% | |
| 99% | [latex]\pm[/latex]6% |
Hint: Use the sliders to set the given values.
Question 7
Now uncheck the “Use Conservative Approach” box. Use the same tool to calculate the necessary sample sizes for the following scenarios.
| Confidence Level | Margin of Error | Population
Proportion |
Minimum Sample Size |
| 95% | [latex]\pm[/latex]2.5% | 0.1 | |
| 95% | [latex]\pm[/latex]4% | 0.2 | |
| 95% | [latex]\pm[/latex]6% | 0.3 | |
| 99% | [latex]\pm[/latex]6% | 0.3 |
Hint: Use the sliders to set the given values.
Notice that using the conservative [latex]\hat{p} = 0.5[/latex] approach always yields a larger than necessary sample size.
Looking Ahead
Choose a context that is interesting to you, and bring the following information with you to class.
Question 8
Write a question that can be answered using proportions.
Hint: Remember that percentages are ways to express information about proportions.
Question 9
Use an Internet search to find an answer for your question. Note the source and any additional information provided regarding sample size, margin of error, or confidence level.
Hint: Ask a librarian if you need help with your Internet search.
