Answer these questions **Understanding Confidence Intervals: An Educational Guide****Introduction**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.**Problem Statement**From a random sample of 65 dates, the mean record high daily temperature in a certain city has a mean of 83.83°F. Assume the population standard deviation is 14.13°F.**Tasks**1. The 90% confidence interval is (______ , ______) *(Round to two decimal places as needed.)*2. The 95% confidence interval is (______ , ______) *(Round to two decimal places as needed.)*3. Which interval is wider? Choose the correct answer below. - The 90% confidence interval - The 95% confidence interval**Explanation**- **Confidence Interval (CI)**: A range of values derived from sample statistics that could contain the population parameter (mean) with a specified level of confidence (e.g., 90% or 95%).- **Sample Size (n)**: The number of observations in the sample, which in this case is 65.- **Sample Mean (x̄)**: The average value from the sample. For this problem, it is 83.83°F.- **Population Standard Deviation (σ)**: A measure of the amount of variation or dispersion in the population, given as 14.13°F.**Confidence Interval Calculation**1. **Determine the critical value (z) for the desired level of confidence**: - For a 90% CI, \( z \) is approximately 1.645. - For a 95% CI, \( z \) is approximately 1.96.2. **Calculate the standard error of the mean (SE)**: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{14.13}{\sqrt{65}} \]3. **Construct the Confidence Interval using the formula**: \[ \text{CI} = \bar{x} \pm z \cdot SE \]**Interpreting the Results**The wider interval between the two options (90% CI and 95% CI) will offer more certainty that the population mean lies within that range. ## Understanding Confidence Intervals: Educational Guide### Scenario:You are provided with the sample mean and the population standard deviation. This information will be used to construct the 90% and 95% confidence intervals for the population mean. You will interpret the results and compare the widths of the confidence intervals.#### Given Data:- **Sample Size**: 65 dates- **Sample Mean High Daily Temperature**: 83.83°F- **Population Standard Deviation**: 14.13°F---### Task:Interpret the results. #### Multiple Choice Options:**A.** You can be 90% confident that the population mean record high temperature is between the bounds of the 90% confidence interval, and 95% confident for the 95% interval.**B.** You can be 90% confident that the population mean record high temperature is outside the bounds of the 90% confidence interval, and 95% confident for the 95% interval.**C.** You can be certain that the mean record high temperature was within the 90% confidence interval for approximately 59 of the 65 days, and was within the 95% confidence interval for approximately 62 of the 65 days.**D.** You can be certain that the population mean record high temperature is either between the lower bounds of the 90% and 95% confidence intervals or the upper bounds of the 90% and 95% confidence intervals.### Explanation:- **Confidence Interval:** Indicates a range of values, derived from the sample data, that is likely to contain the value of an unknown population parameter. The confidence level represents the frequency (percentage) of possible confidence intervals that contain the true population parameter.- **90% Confidence Interval:** We are 90% confident that the interval captures the true population mean. There is a 10% chance that the true mean falls outside this interval.- **95% Confidence Interval:** We are 95% confident that the interval captures the true population mean. There is a 5% chance that the true mean falls outside this interval.### Interpretation:- **Option A** is the correct interpretation. It correctly states that there is a 90% confidence for the 90% interval and a 95% confidence for the 95% interval.- **Option B** is incorrect as it incorrectly states the confidence outside the interval, which contradicts the confidence interval concept.- **

It is given that Sample mean = 83.83Population standard deviation = 14.13Sample size = 65

The 90% confidence interval is (). (Round to two decimal places as needed.) The 95% confidence interval is (). (Round to two decimal places as needed.) Which interval is wider? Choose the correct answer below. The 90% confidence interval The 95% confidence interval Interpret the results.

The 90% confidence interval is (). (Round to two decimal places as needed.) The 95% confidence interval is (). (Round to two decimal places as needed.) Which interval is wider? Choose the correct answer below. The 90% confidence interval The 95% confidence interval Interpret the results.

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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Question

Answer these questions

**Understanding Confidence Intervals: An Educational Guide**

**Introduction**

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.

**Problem Statement**

From a random sample of 65 dates, the mean record high daily temperature in a certain city has a mean of 83.83°F. Assume the population standard deviation is 14.13°F.

**Tasks**

1. The 90% confidence interval is (______ , ______)
*(Round to two decimal places as needed.)*

2. The 95% confidence interval is (______ , ______)
*(Round to two decimal places as needed.)*

3. Which interval is wider? Choose the correct answer below.
- The 90% confidence interval
- The 95% confidence interval

**Explanation**

- **Confidence Interval (CI)**: A range of values derived from sample statistics that could contain the population parameter (mean) with a specified level of confidence (e.g., 90% or 95%).

- **Sample Size (n)**: The number of observations in the sample, which in this case is 65.

- **Sample Mean (x̄)**: The average value from the sample. For this problem, it is 83.83°F.

- **Population Standard Deviation (σ)**: A measure of the amount of variation or dispersion in the population, given as 14.13°F.

**Confidence Interval Calculation**

1. **Determine the critical value (z) for the desired level of confidence**:
- For a 90% CI, \( z \) is approximately 1.645.
- For a 95% CI, \( z \) is approximately 1.96.

2. **Calculate the standard error of the mean (SE)**:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{14.13}{\sqrt{65}}
\]

3. **Construct the Confidence Interval using the formula**:
\[
\text{CI} = \bar{x} \pm z \cdot SE
\]

**Interpreting the Results**

The wider interval between the two options (90% CI and 95% CI) will offer more certainty that the population mean lies within that range.

## Understanding Confidence Intervals: Educational Guide

### Scenario:
You are provided with the sample mean and the population standard deviation. This information will be used to construct the 90% and 95% confidence intervals for the population mean. You will interpret the results and compare the widths of the confidence intervals.

#### Given Data:
- **Sample Size**: 65 dates
- **Sample Mean High Daily Temperature**: 83.83°F
- **Population Standard Deviation**: 14.13°F

---

### Task:
Interpret the results.

#### Multiple Choice Options:
**A.** You can be 90% confident that the population mean record high temperature is between the bounds of the 90% confidence interval, and 95% confident for the 95% interval.

**B.** You can be 90% confident that the population mean record high temperature is outside the bounds of the 90% confidence interval, and 95% confident for the 95% interval.

**C.** You can be certain that the mean record high temperature was within the 90% confidence interval for approximately 59 of the 65 days, and was within the 95% confidence interval for approximately 62 of the 65 days.

**D.** You can be certain that the population mean record high temperature is either between the lower bounds of the 90% and 95% confidence intervals or the upper bounds of the 90% and 95% confidence intervals.

### Explanation:
- **Confidence Interval:** Indicates a range of values, derived from the sample data, that is likely to contain the value of an unknown population parameter. The confidence level represents the frequency (percentage) of possible confidence intervals that contain the true population parameter.

- **90% Confidence Interval:** We are 90% confident that the interval captures the true population mean. There is a 10% chance that the true mean falls outside this interval.

- **95% Confidence Interval:** We are 95% confident that the interval captures the true population mean. There is a 5% chance that the true mean falls outside this interval.

### Interpretation:
- **Option A** is the correct interpretation. It correctly states that there is a 90% confidence for the 90% interval and a 95% confidence for the 95% interval.

- **Option B** is incorrect as it incorrectly states the confidence outside the interval, which contradicts the confidence interval concept.

- **

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