Questions: 8. [-/4 Points] DETAILS MY NOTES 0/2 Submissions Used You want to conduct a survey to determine the proportion of people who favor a proposed tax policy. How does increasing the sample size affect the size of the margin of error? As the sample size increases, the margin of error decreases. As the sample size increases, the margin of error remains the same. As the sample size increases, the margin of error increases.

8. [-/4 Points]

DETAILS
MY NOTES
0/2 Submissions Used
You want to conduct a survey to determine the proportion of people who favor a proposed tax policy. How does increasing the sample size affect the size of the margin of error?
As the sample size increases, the margin of error decreases.
As the sample size increases, the margin of error remains the same.
As the sample size increases, the margin of error increases.

Transcript text: 8. [-/4 Points] DETAILS MY NOTES 0/2 Submissions Used You want to conduct a survey to determine the proportion of people who favor a proposed tax policy. How does increasing the sample size affect the size of the margin of error? As the sample size increases, the margin of error decreases. As the sample size increases, the margin of error remains the same. As the sample size increases, the margin of error increases.

Solution

Solution Steps

Step 1: Initial Margin of Error Calculation

To determine the initial margin of error for a sample size of \( n = 100 \), we use the formula:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} \]

Given that the Z-Score \( Z = 1.96 \) and the population standard deviation \( \sigma = 15 \), we can substitute these values into the formula:

\[ \text{Margin of Error} = \frac{1.96 \times 15}{\sqrt{100}} = 2.9399 \]

Thus, the initial margin of error is \( 2.9399 \).

Step 2: New Margin of Error Calculation

Next, we calculate the margin of error for an increased sample size of \( n = 200 \):

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} = \frac{1.96 \times 15}{\sqrt{200}} = 2.0789 \]

The new margin of error is \( 2.0789 \).

Step 3: Further Margin of Error Calculation

Finally, we calculate the margin of error for a further increased sample size of \( n = 400 \):

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} = \frac{1.96 \times 15}{\sqrt{400}} = 1.47 \]

The further new margin of error is \( 1.47 \).

Final Answer

As the sample size increases from \( 100 \) to \( 200 \) and then to \( 400 \), the margin of error decreases from \( 2.9399 \) to \( 2.0789 \) and finally to \( 1.47 \).

Thus, the conclusion is that as the sample size increases, the margin of error decreases.

The answer is \( \boxed{\text{As the sample size increases, the margin of error decreases.}} \)

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