12 Volunteers were randomly divided into two equal groups. Group A subjects were tested
sensory deprivation, while group B individuals served as witnesses. At the end of the test.
in the experiment, the alpha wave frequency component of the subject electroencephalograms was measured. The results were as follows:
Group A: 10.2, 9.5, 10.1, 10.0, 9.8, 10.9; Group B: 11.0, 11.2, 10.1, 11.4, 11.7, 11.2
a) A statistically significant difference (a=0.05) is documented in the mean values of the EEG wave frequency component between the two populations;
b) Calculate the 95% confidence interval of the difference (Group A minus Group B) of the average frequency component of alpha waves, the, under the assumption that the two populations are normally distributed and have equal fluctuations.


We are tasked with comparing the means of two groups (Group A and Group B) to determine if there is a statistically significant difference in their alpha wave frequency components. To achieve this, we'll use a **two-sample t-test** under the assumption of equal variances, since it's mentioned that the populations have equal fluctuations. 

We'll then calculate the **95% confidence interval** for the difference in the mean values between the two groups.

### Steps to Solve:

#### Part a) Test for Statistical Significance (Two-Sample t-test)

1. **Null hypothesis (\(H_0\))**: The means of the two groups are equal.
   \[
   H_0: \mu_A = \mu_B
   \]

2. **Alternative hypothesis (\(H_1\))**: The means of the two groups are different.
   \[
   H_1: \mu_A \neq \mu_B
   \]

3. **Calculate the sample means and variances** for Group A and Group B.
   - Group A: \(10.2, 9.5, 10.1, 10.0, 9.8, 10.9\)
   - Group B: \(11.0, 11.2, 10.1, 11.4, 11.7, 11.2\)

4. **Pooled standard deviation**:
   Since we're assuming equal variances, we'll use the pooled variance formula.

5. **t-statistic**:
   \[
   t = \frac{\bar{x}_A - \bar{x}_B}{\text{standard error of the difference}}
   \]

6. Compare the t-statistic to the critical value from the t-distribution with the appropriate degrees of freedom to check if the result is statistically significant at \( \alpha = 0.05 \).

#### Part b) 95% Confidence Interval for the Difference

To calculate the **95% confidence interval** for the difference in means (\( \mu_A - \mu_B \)):
\[
CI = \left( \bar{x}_A - \bar{x}_B \right) \pm t_{\alpha/2} \times \text{standard error of the difference}
\]
Where \(t_{\alpha/2}\) is the critical t-value for 95% confidence and the appropriate degrees of freedom.

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Let's calculate the necessary values now.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through how to do the calculations yourself or answer any other questions you might have.

Here are five follow-up questions you could explore:
1. What assumptions must hold true for the two-sample t-test to be valid?
2. How would you calculate the confidence interval if the two populations had unequal variances?
3. How is the pooled variance calculated in a two-sample t-test?
4. What is the interpretation of the p-value in hypothesis testing?
5. How does changing the confidence level (e.g., from 95% to 99%) affect the confidence interval?

**Tip**: Always check the normality of the data and equality of variances before applying a t-test to ensure accurate results.

12 Volunteers were randomly divided into two equal groups. Group A subjects were tested in sensory deprivation, while Group B individuals served as witnesses. At the end of the experiment, the alpha wave frequency component of the subject electroencephalograms was measured. The results were as follows: Group A: 10.2, 9.5, 10.1, 10.0, 9.8, 10.9; Group B: 11.0, 11.2, 10.1, 11.4, 11.7, 11.2. a) Is a statistically significant difference (a=0.05) documented in the mean values of the EEG wave frequency component between the two populations? b) Calculate the 95% confidence interval of the difference (Group A minus Group B) of the average frequency component of alpha waves, under the assumption that the two populations are normally distributed and have equal fluctuations.

Explore a detailed solution for comparing the alpha wave frequency components in sensory deprivation using a two-sample t-test. This example covers hypothesis testing, confidence intervals, and the pooled variance formula. Suitable for college-level statistics students.

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