### Constructing a 95% Confidence Interval for the Population MeanWhen estimating the population mean (μ) from sample data, it's crucial to construct a confidence interval to understand the range in which the true population mean is likely to fall. In this example, we assume the population has a normal distribution. A sample of 15 randomly selected students has provided the following data:- **Sample Mean (x̄):** 2.39- **Standard Deviation (s):** 0.83Given this information, we aim to construct a 95% confidence interval for the population mean.#### Options Provided for the Confidence Interval:1. $ (1.65, 4.05) $2. $ (1.93, 2.85) $3. $ (2.35, 3.35) $4. $ (2.75, 2.95) $You are required to select the correct interval from these options.### Understanding Confidence IntervalsA 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 95 of those intervals would contain the population mean μ.#### Calculation Details (Conceptual):1. **Determine the Critical Value:** For a 95% confidence interval in a two-tailed test, the critical value (z*) for a normal distribution is approximately 1.96. 2. **Calculate the Standard Error (SE):** The standard error is given by the formula: \[ SE = \frac{s}{\sqrt{n}} \] where $s$ is the standard deviation and $n$ is the sample size.3. **Compute the Margin of Error (ME):** Margin of error is computed as: \[ ME = z* \times SE \]4. **Construct the Confidence Interval:** The confidence interval is then defined as: \[ CI = \left( \bar{x} - ME, \bar{x} + ME \right) \]### Instructions:- Carefully use the provided data to calculate or validate the confidence interval.- Select the correct interval from the options provided. Please proceed by selecting the interval that you determine to be correct.#### Interactive Element:Click the button labeled "Continue" once you have made your selection.*Continue [Button

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Construct a 95% confidence interval for the population mean, u. Assume the population has a normal distribution. A sample of 15 randomly selected students has a grade point average of 2.39 with a standard deviation of 0.83. O(1.65, 4.05) o (1.93, 2.85) O (2.35, 3.35) O (2.75, 2.95)

Construct a 95% confidence interval for the population mean, u. Assume the population has a normal distribution. A sample of 15 randomly selected students has a grade point average of 2.39 with a standard deviation of 0.83. O(1.65, 4.05) o (1.93, 2.85) O (2.35, 3.35) O (2.75, 2.95)

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

### Constructing a 95% Confidence Interval for the Population Mean

When estimating the population mean (μ) from sample data, it's crucial to construct a confidence interval to understand the range in which the true population mean is likely to fall.

In this example, we assume the population has a normal distribution. A sample of 15 randomly selected students has provided the following data:

- **Sample Mean (x̄):** 2.39
- **Standard Deviation (s):** 0.83

Given this information, we aim to construct a 95% confidence interval for the population mean.

#### Options Provided for the Confidence Interval:

1. $ (1.65, 4.05) $
2. $ (1.93, 2.85) $
3. $ (2.35, 3.35) $
4. $ (2.75, 2.95) $

You are required to select the correct interval from these options.

### Understanding Confidence Intervals

A 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 95 of those intervals would contain the population mean μ.

#### Calculation Details (Conceptual):

1. **Determine the Critical Value:** For a 95% confidence interval in a two-tailed test, the critical value (z*) for a normal distribution is approximately 1.96.

2. **Calculate the Standard Error (SE):**
The standard error is given by the formula:
\[
SE = \frac{s}{\sqrt{n}}
\]
where $s$ is the standard deviation and $n$ is the sample size.

3. **Compute the Margin of Error (ME):**
Margin of error is computed as:
\[
ME = z* \times SE
\]

4. **Construct the Confidence Interval:**
The confidence interval is then defined as:
\[
CI = \left( \bar{x} - ME, \bar{x} + ME \right)
\]

### Instructions:

- Carefully use the provided data to calculate or validate the confidence interval.
- Select the correct interval from the options provided.

Please proceed by selecting the interval that you determine to be correct.

#### Interactive Element:
Click the button labeled "Continue" once you have made your selection.

*Continue [Button

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