**Comparative Study of Graduate and Undergraduate Student Grades****Overview:**This study involves comparing the grades of two independently selected groups of students: graduates and undergraduates. The grades are assumed to follow a normal distribution with unknown parameters.**Data Summary Table:**| Group | Size | Mean ($\bar{X}$) | Variance ($S^2$) | $v = S^2/n$ ||----------------|------|---------------------|--------------------|---------------|| Graduate | $n_1 = 10$ | $(\bar{X})_1 = 85.1$ | $(S^2)_1 = 40$ | $v_1 = \frac{40}{10} = 4$ || Undergraduate | $n_2 = 10$ | $(\bar{X})_2 = 70.9$ | $(S^2)_2 = 600$ | $v_2 = \frac{600}{10} = 60$ |**Hypothesis Testing Framework:**For all hypothesis testing scenarios related to this study, follow the standardized conclusion format:1. **Test Statistic Value:** Calculate the value of the test statistic.2. **Critical Values Required:** Determine the critical values for the chosen level of significance.3. **State Rejection Rule:** Clearly explain the decision-making criteria, whether to reject or not reject the null hypothesis.4. **Decision:** Articulate the conclusion, such as "yes, reject the null" or "not enough evidence for rejection."**Confidence Intervals:**To derive confidence intervals, provide the critical values required and present the interval using the following format:- Upper Confidence Limit (UCL) = - Lower Confidence Limit (LCL) = This structured approach ensures clarity and consistency in hypothesis testing and confidence interval derivation. **Text Transcription for Educational Website:**---Continue with the same summaries and assumption as in Problem 3. Faculty assumed that two populations have **ENTIRELY UNKNOWN VARIANCES**, $(\sigma_1)^2$ and $(\sigma_2)^2$. Samples of size $ n_1 = n_2 = n = 10 $ were collected and summarized, with summaries presented in the table shown on page 2.Estimate the difference, $ \mu = \mu_1 - \mu_2 $ between two population means with confidence $ C = 0.95 $.1. Show critical value (or values) needed for this procedure.2. Evaluate upper and lower confidence limits for the targeted parameter, $ \mu_1 - \mu_2 $.--- *Note: Since no graphs or diagrams are provided in the image, there are none to describe.*

Answered: Problems 1 - 4 share the same summaries…

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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**Comparative Study of Graduate and Undergraduate Student Grades**

**Overview:**
This study involves comparing the grades of two independently selected groups of students: graduates and undergraduates. The grades are assumed to follow a normal distribution with unknown parameters.

**Data Summary Table:**

| Group | Size | Mean ($\bar{X}$) | Variance ($S^2$) | $v = S^2/n$ |
|----------------|------|---------------------|--------------------|---------------|
| Graduate | $n_1 = 10$ | $(\bar{X})_1 = 85.1$ | $(S^2)_1 = 40$ | $v_1 = \frac{40}{10} = 4$ |
| Undergraduate | $n_2 = 10$ | $(\bar{X})_2 = 70.9$ | $(S^2)_2 = 600$ | $v_2 = \frac{600}{10} = 60$ |

**Hypothesis Testing Framework:**

For all hypothesis testing scenarios related to this study, follow the standardized conclusion format:

1. **Test Statistic Value:** Calculate the value of the test statistic.
2. **Critical Values Required:** Determine the critical values for the chosen level of significance.
3. **State Rejection Rule:** Clearly explain the decision-making criteria, whether to reject or not reject the null hypothesis.
4. **Decision:** Articulate the conclusion, such as "yes, reject the null" or "not enough evidence for rejection."

**Confidence Intervals:**

To derive confidence intervals, provide the critical values required and present the interval using the following format:

- Upper Confidence Limit (UCL) =
- Lower Confidence Limit (LCL) =

This structured approach ensures clarity and consistency in hypothesis testing and confidence interval derivation.

**Text Transcription for Educational Website:**

---

Continue with the same summaries and assumption as in Problem 3. Faculty assumed that two populations have **ENTIRELY UNKNOWN VARIANCES**, $(\sigma_1)^2$ and $(\sigma_2)^2$. Samples of size $ n_1 = n_2 = n = 10 $ were collected and summarized, with summaries presented in the table shown on page 2.

Estimate the difference, $ \mu = \mu_1 - \mu_2 $ between two population means with confidence $ C = 0.95 $.

1. Show critical value (or values) needed for this procedure.
2. Evaluate upper and lower confidence limits for the targeted parameter, $ \mu_1 - \mu_2 $.

---

*Note: Since no graphs or diagrams are provided in the image, there are none to describe.*

Expert Solution

Step 1

Hey, since there are multiple subparts posted, we will answer first three question. If you want any specific question to be answered then please submit that question only or specify the question number in your message.

The test statistic is 1.775, which is obtained below:

$\begin{array}{rcl} t & = & \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}} \\ = & \frac{85.1 - 70.9}{\sqrt{4 + 60}} \\ = & 1.775 \end{array}$

The degrees of freedom is,

$\begin{array}{rcl} d f & = & s m a l l e r o f (n 1 - 1) (n 2 - 1) \\ = & 10 - 1 \\ = & 9 \end{array}$

The critical-value is obtained by using Excel function “=TINV(probability, degrees of freedom)”.

Output obtained from Excel is given below:

From the output, the critical value for two tailed test with 0.05 level is 2.262.

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