A survey asked, "How important is it to you to buy products that are made in America?" of the 600 Americans surveyed, 450 responded, "It is important." For each of the following levels of confidence, find the sample proportion and the margin of error (MOE) associated with the poll. (Give the answers to one decimal place, if needed.)
(a) A 90% level of confidence
Sample proportion is
75

Correct: Your answer is correct.
%
MOE =

Incorrect: Your answer is incorrect.
%

(b) A 95% level of confidence
Sample proportion is
75

Correct: Your answer is correct.
%
MOE =

Incorrect: Your answer is incorrect.
%

Let's work through the problem step-by-step for each confidence level. The key elements we need to find are:

1. **Sample proportion (\( \hat{p} \))**: This is the proportion of survey respondents who said it is important to buy products made in America. It is calculated as:
   \[
   \hat{p} = \frac{\text{Number of "important" responses}}{\text{Total number of responses}} = \frac{450}{600} = 0.75
   \]

2. **Margin of Error (MOE)**: This is calculated using the formula:
   \[
   \text{MOE} = Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
   \]
   Where:
   - \( Z_{\alpha/2} \) is the critical value corresponding to the confidence level.
   - \( \hat{p} \) is the sample proportion.
   - \( n \) is the sample size (in this case, 600).

Let's calculate for both (a) 90% and (b) 95% confidence levels.

### (a) A 90% Level of Confidence

1. **Sample Proportion**: We already have the sample proportion, \( \hat{p} = 0.75 \) (or 75%).

2. **Critical Value (\( Z_{\alpha/2} \))** for 90% confidence: The critical value for a 90% confidence level is approximately 1.645.

3. **Margin of Error (MOE)**:
   \[
   \text{MOE} = 1.645 \times \sqrt{\frac{0.75(1 - 0.75)}{600}} = 1.645 \times \sqrt{\frac{0.75 \times 0.25}{600}}
   \]
   Let's calculate the margin of error.

### (b) A 95% Level of Confidence

1. **Sample Proportion**: \( \hat{p} = 0.75 \) (or 75%).

2. **Critical Value (\( Z_{\alpha/2} \))** for 95% confidence: The critical value for a 95% confidence level is approximately 1.96.

3. **Margin of Error (MOE)**:
   \[
   \text{MOE} = 1.96 \times \sqrt{\frac{0.75(1 - 0.75)}{600}} = 1.96 \times \sqrt{\frac{0.75 \times 0.25}{600}}
   \]
   Let's calculate this margin of error as well.

I'll calculate these margins of error now.Here are the results:

### (a) A 90% Level of Confidence:
- Sample proportion: \( 75\% \)
- Margin of error (MOE): \( 2.9\% \)

### (b) A 95% Level of Confidence:
- Sample proportion: \( 75\% \)
- Margin of error (MOE): \( 3.5\% \)

Would you like further details on the calculations, or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How would the margin of error change if the sample size was larger?
2. What is the relationship between confidence level and margin of error?
3. How is the critical value (\( Z_{\alpha/2} \)) chosen for different confidence levels?
4. What would happen to the margin of error if the proportion of positive responses was closer to 50%?
5. How does increasing the confidence level affect the width of a confidence interval?

**Tip**: Increasing the sample size reduces the margin of error, providing more precise results.

A survey asked, 'How important is it to you to buy products that are made in America?' Of the 600 Americans surveyed, 450 responded, 'It is important.' For each of the following levels of confidence, find the sample proportion and the margin of error (MOE) associated with the poll. (a) A 90% level of confidence, (b) A 95% level of confidence.

Learn how to calculate the sample proportion and margin of error for a survey asking about the importance of buying American-made products. This step-by-step guide covers confidence intervals, critical values, and sample proportions, with calculations for both 90% and 95% confidence levels. Suitable for high school students in Grades 10-12.

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