You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 68 dates, the mean record high daily temperature in a certain city has a mean of 36.86°F Assume the population standard deviation is 13.66°F The 90% confidence interval is (). Round to two decimal places as needed.) GERM
You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 68 dates, the mean record high daily temperature in a certain city has a mean of 36.86°F Assume the population standard deviation is 13.66°F The 90% confidence interval is (). Round to two decimal places as needed.) GERM
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Constructing Confidence Intervals for Population Mean**
In this exercise, you are given the sample mean and the population standard deviation. You are tasked with using this information to construct the 90% and 95% confidence intervals for the population mean. You will then interpret the results and compare the widths of the confidence intervals.
### Given Data:
- **Sample Size (n):** 68
- **Sample Mean (x̄):** 86.86°F
- **Population Standard Deviation (σ):** 13.66°F
Using this information, you need to construct the 90% confidence interval for the population mean.
### Steps to Construct the 90% Confidence Interval:
1. **Determine the Z-Score for 90% Confidence Level:**
For a 90% confidence level, the critical value (Z*) is typically 1.645 (you can find this in standard Z-tables).
2. **Calculating the Standard Error of the Mean (SEM):**
\[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{13.66}{\sqrt{68}}
\]
3. **Calculate the Margin of Error (ME):**
\[
ME = Z^* \times SEM
\]
4. **Construct the Confidence Interval:**
\[
\text{Lower Limit} = x̄ - ME
\]
\[
\text{Upper Limit} = x̄ + ME
\]
This will give you the 90% confidence interval.
### Interpretation:
- **Confidence Interval Range:** [Lower Limit, Upper Limit]
- Compare the widths of the confidence intervals for different confidence levels (e.g., 90% vs 95%) to understand how the interval width changes with confidence level.
### Example Calculation for 90% Confidence Interval:
The 90% confidence interval is [____ , ____]. (Round to two decimal places as needed.)
By following these steps, you can construct and interpret the confidence intervals for various confidence levels. This exercise helps in understanding how confident we can be about the population mean based on sample data.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F952bcd3e-4a7c-474a-8dd6-ddd08471460a%2F342ce26f-08c8-4a2c-abbe-d4b42b571a84%2Fd4ihdbm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Constructing Confidence Intervals for Population Mean**
In this exercise, you are given the sample mean and the population standard deviation. You are tasked with using this information to construct the 90% and 95% confidence intervals for the population mean. You will then interpret the results and compare the widths of the confidence intervals.
### Given Data:
- **Sample Size (n):** 68
- **Sample Mean (x̄):** 86.86°F
- **Population Standard Deviation (σ):** 13.66°F
Using this information, you need to construct the 90% confidence interval for the population mean.
### Steps to Construct the 90% Confidence Interval:
1. **Determine the Z-Score for 90% Confidence Level:**
For a 90% confidence level, the critical value (Z*) is typically 1.645 (you can find this in standard Z-tables).
2. **Calculating the Standard Error of the Mean (SEM):**
\[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{13.66}{\sqrt{68}}
\]
3. **Calculate the Margin of Error (ME):**
\[
ME = Z^* \times SEM
\]
4. **Construct the Confidence Interval:**
\[
\text{Lower Limit} = x̄ - ME
\]
\[
\text{Upper Limit} = x̄ + ME
\]
This will give you the 90% confidence interval.
### Interpretation:
- **Confidence Interval Range:** [Lower Limit, Upper Limit]
- Compare the widths of the confidence intervals for different confidence levels (e.g., 90% vs 95%) to understand how the interval width changes with confidence level.
### Example Calculation for 90% Confidence Interval:
The 90% confidence interval is [____ , ____]. (Round to two decimal places as needed.)
By following these steps, you can construct and interpret the confidence intervals for various confidence levels. This exercise helps in understanding how confident we can be about the population mean based on sample data.
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