Problems 1 - 4 share the same summaries presented in the table below. Two groups of students, graduates and undergraduates, were subject to comparative studies. Their individual grades are assumed to be normal with unknown parameters. Respondents were selected independently and their sample summaries are presented in the table below. Group Size Mean = X Variance = S2 v = S² /n Graduate nį = 10 | (X)1 = 85.1 (S²)1 = 40 v1 = 10 = 4 Undergraduate n2 = 10 (X)2 = 70.9 (S2)2 = 600 V2 = = 60 For all hypothesis testing problems, you are supposed to show your conclusions in the standardized form as follows. 1. Test statistic value 2. Critical values required 3. State rejection rule explaining what you decide 4. The decision such as "yes, reject the null" or "not enough evidence for rejection" Deriving confidence intervals, please show critical values needed and present your interval in the form UCL = LCL =

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**Problems 1 - 4: Summary and Hypothesis Testing Instructions**

Two groups of students, graduates and undergraduates, were subject to a comparative study. Their individual grades are assumed to be normally distributed with unknown parameters. The sample summaries for each group are as follows:

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

### Instructions for Hypothesis Testing Problems

When approaching hypothesis testing problems, present your conclusions in a standardized format as follows:

1. **Test Statistic Value**: Calculate and state the test statistic value.
2. **Critical Values Required**: Determine and specify the critical values needed for comparison.
3. **State Rejection Rule**: Explain the rejection rule and your decision-making criteria.
4. **Decision**: Conclude with a statement such as "yes, reject the null" or "not enough evidence for rejection."

### Confidence Intervals

To derive confidence intervals, determine the critical values needed and present your interval in the form:

- **UCL =** [Upper Confidence Limit]
- **LCL =** [Lower Confidence Limit]
Transcribed Image Text:**Problems 1 - 4: Summary and Hypothesis Testing Instructions** Two groups of students, graduates and undergraduates, were subject to a comparative study. Their individual grades are assumed to be normally distributed with unknown parameters. The sample summaries for each group are as follows: | **Group** | **Size** | **Mean = \(\bar{X}\)** | **Variance = \(S^2\)** | \(v = \frac{S^2}{n}\) | |-------------------|----------|------------------------|------------------------|-----------------------| | **Graduate** | \(n_1 = 10\) | \((\bar{X})_1 = 85.1\) | \((S^2)_1 = 40\) | \(v_1 = \frac{40}{10} = 4\) | | **Undergraduate** | \(n_2 = 10\) | \((\bar{X})_2 = 70.9\) | \((S^2)_2 = 600\) | \(v_2 = \frac{600}{10} = 60\) | ### Instructions for Hypothesis Testing Problems When approaching hypothesis testing problems, present your conclusions in a standardized format as follows: 1. **Test Statistic Value**: Calculate and state the test statistic value. 2. **Critical Values Required**: Determine and specify the critical values needed for comparison. 3. **State Rejection Rule**: Explain the rejection rule and your decision-making criteria. 4. **Decision**: Conclude with a statement such as "yes, reject the null" or "not enough evidence for rejection." ### Confidence Intervals To derive confidence intervals, determine the critical values needed and present your interval in the form: - **UCL =** [Upper Confidence Limit] - **LCL =** [Lower Confidence Limit]
**Statistical Analysis of Population Means with Unknown Variances**

Faculty assumed that two populations have **ENTIRELY UNKNOWN VARIANCES**, denoted as \((\sigma_1)^2\) 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. The objective is to draw conclusions about \( \mu = \mu_1 - \mu_2 \), the difference of two population means. At the significance level, \( \alpha = 0.05 \), do you have evidence that the graduates had a higher population average grade than undergrads?

1. **Show critical value with indication of the source you got it**
   
2. **State rejection rule**

3. **Evaluate the test statistic and formulate your decision**

---

*Note*: Be sure to consult appropriate statistical tables, such as the t-distribution table, to find critical values based on your specific significance level, sample size, and test type. Consider using software or online tools to compute test statistics and make data-driven conclusions effectively.
Transcribed Image Text:**Statistical Analysis of Population Means with Unknown Variances** Faculty assumed that two populations have **ENTIRELY UNKNOWN VARIANCES**, denoted as \((\sigma_1)^2\) 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. The objective is to draw conclusions about \( \mu = \mu_1 - \mu_2 \), the difference of two population means. At the significance level, \( \alpha = 0.05 \), do you have evidence that the graduates had a higher population average grade than undergrads? 1. **Show critical value with indication of the source you got it** 2. **State rejection rule** 3. **Evaluate the test statistic and formulate your decision** --- *Note*: Be sure to consult appropriate statistical tables, such as the t-distribution table, to find critical values based on your specific significance level, sample size, and test type. Consider using software or online tools to compute test statistics and make data-driven conclusions effectively.
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