Let's walk through both parts of the problem.

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(a) Margin of Error Calculation (95% Confidence Level)

The formula for the margin of error (ME) for a sample mean is:

$ME = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

where:

• $Z_{\alpha/2}$ is the critical value for a 95% confidence level (1.960 for 95% confidence),
• $\sigma$ is the standard deviation of the sample (7 pounds),
• $n$ is the sample size (121 bags).

Now, substitute the values:

$ME = 1.960 \times \frac{7}{\sqrt{121}}$

$ME = 1.960 \times \frac{7}{11} = 1.960 \times 0.6364 \approx 1.2467 \, \text{pounds}$

So, the margin of error is:

$\boxed{1.2467 \, \text{pounds}}$

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Explanation of the Margin of Error:

Out of the provided options, the correct interpretation of the margin of error at 95% confidence is:

"We can say with 0.95 probability that the distance between the sample mean of 37 pounds and the population mean is at most the margin of error calculated above."

This means that with 95% confidence, the true population mean is within the range determined by adding and subtracting the margin of error from the sample mean.

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(b) 95% Confidence Interval for the Population Mean

The formula for the confidence interval is:

$\text{CI} = \bar{X} \pm ME$

where:

• $\bar{X} = 37$ pounds (sample mean),
• $ME = 1.2467$ pounds (calculated above).

Now, calculate the lower and upper bounds of the interval:

$\text{Lower bound} = 37 - 1.2467 = 35.75$ $\text{Upper bound} = 37 + 1.2467 = 38.25$

So, the 95% confidence interval for the population mean is:

$\boxed{(35.75, 38.25)} \, \text{pounds}$

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Summary

• (a) Margin of error: 1.2467 pounds
• (b) 95% confidence interval: (35.75, 38.25) pounds

Would you like further details on any part of the calculation?

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Here are 5 related questions for deeper exploration:

1. How would the margin of error change if the sample size increased to 400?
2. What is the Z-critical value for a 99% confidence level, and how does it affect the margin of error?
3. How do we interpret a confidence interval that includes the value 0?
4. If the standard deviation was 10 pounds instead of 7, how would the margin of error change?
5. What assumptions must hold for the confidence interval to be valid?

Tip: Increasing the sample size decreases the margin of error, making the estimate more precise.