In a sample of 57 temperature readings taken from the freezer of a restaurant, the mean is 29.6 degrees and the population standard deviation is 2.7 degrees. What would be the 80% confidence interval for the temperatures in the freezer?

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5VA. Confidence intervals with population standard deviationLinks to an external site. (1:42)

Group of answer choices

(26.91, 32.31)

(29.14, 30.06)

(31.90, 32.44)

(24.25, 35.05)

To calculate the 80% confidence interval for the temperatures in the freezer, we use the formula for a confidence interval when the population standard deviation is known:

\[ \text{CI} = \bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right) \]

Where:
- \(\bar{x}\) is the sample mean
- \(z_{\alpha/2}\) is the z-score corresponding to the desired confidence level
- \(\sigma\) is the population standard deviation
- \(n\) is the sample size

Given:
- \(\bar{x} = 29.6\)
- \(\sigma = 2.7\)
- \(n = 57\)

For an 80% confidence interval, the z-score \(z_{\alpha/2}\) corresponding to a confidence level of 80% (which leaves 10% in each tail of the normal distribution) is approximately 1.28.

Now, let's calculate the margin of error (ME):

\[ \text{ME} = z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right) \]

\[ \text{ME} = 1.28 \left(\frac{2.7}{\sqrt{57}}\right) \]

\[ \text{ME} = 1.28 \left(\frac{2.7}{7.55}\right) \]

\[ \text{ME} = 1.28 \left(0.3574\right) \]

\[ \text{ME} \approx 0.457 \]

Next, we calculate the lower and upper bounds of the confidence interval:

\[ \text{Lower bound} = \bar{x} - \text{ME} = 29.6 - 0.457 \approx 29.14 \]

\[ \text{Upper bound} = \bar{x} + \text{ME} = 29.6 + 0.457 \approx 30.06 \]

Thus, the 80% confidence interval for the temperatures in the freezer is approximately (29.14, 30.06).

### Group of answer choices:
- (29.14, 30.06)

Would you like more details or have any questions about this conclusion?

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Here are some additional questions that expand on the information provided:

1. How does the confidence level affect the width of the confidence interval?
2. What assumptions must be met to use this method for calculating confidence intervals?
3. How would the confidence interval change if the sample size were larger?
4. What is the z-score for a 90% confidence interval?
5. How would the confidence interval change if the standard deviation were larger?
6. How can you interpret the meaning of this confidence interval in a real-world context?
7. What steps would you take if the sample mean were outside the expected range?
8. How does the population standard deviation impact the reliability of the confidence interval?

**Tip:** When determining confidence intervals, always ensure that the sample data is representative of the population to increase the accuracy of your interval estimates.

Learn how to calculate an 80% confidence interval for temperature readings using the population standard deviation. This problem involves statistics and normal distribution concepts, suitable for high school students.

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