An observational study of Alzheimer's disease (AD) obtained data from 15 AD patients exhibiting moderate dementia and selected a group of 10 control individuals without AD. AD is a progressive neurodegenerative disease of the elderly and advancing age is known to be a primary risk factor in AD diagnosis. Therefore, it was crucial for the study's credibility to examine whether the ages in the AD group might be significantly different than in the control group. The ages of the subjects in years are summarized in the R Output below.
Variable	n	Mean	sd	Minimum	Q1	Median	Q3	Maximum
Alzheimers	15	74.29	6.62	77.00	79.25	87.00	92.25	93.00
Control	10	  65.37	  12.94	    54.00	  56.00	65.00	82.00	89.00
We want to test if the average age in the Alzheimer's group is significantly different than the control group.

(b)	Find the value of the test statistic. in 3 decimals
(c)	Find the 5% critical value. in 3 decimals


To determine whether the average age in the Alzheimer's group is significantly different from that in the control group, we can use an independent samples \( t \)-test. Let’s go through each required step:

### Given Data
For Alzheimer's group:
- \( n_1 = 15 \)
- Mean (\( \bar{X}_1 \)) = 74.29
- Standard deviation (\( s_1 \)) = 6.62

For Control group:
- \( n_2 = 10 \)
- Mean (\( \bar{X}_2 \)) = 65.37
- Standard deviation (\( s_2 \)) = 12.94

### (b) Value of the Test Statistic

1. **Calculate the pooled standard deviation**:
   \[
   s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}
   \]
   
2. **Calculate the test statistic** using the formula for the difference in means:
   \[
   t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
   \]

### (c) Critical Value at 5% Level

Since this is a two-tailed test (testing for a difference in means), the 5% level critical value depends on the degrees of freedom, calculated as:
\[
\text{df} = n_1 + n_2 - 2 = 15 + 10 - 2 = 23
\]

We need to find the \( t \)-value for a two-tailed test with \( \alpha = 0.05 \) and \( \text{df} = 23 \).

Let's calculate these values.The results for the independent samples \( t \)-test are as follows:

- **(b) Test Statistic (t)**: \( 2.276 \)
- **(c) 5% Critical Value**: \( 2.069 \)

These values allow us to assess whether the observed difference in mean ages between the Alzheimer's and control groups is statistically significant.

Would you like to proceed with interpreting the test result? Or do you have any questions?

### Related Questions:
1. How does the critical value change with a larger or smaller sample size?
2. What does the test statistic signify in relation to the null hypothesis?
3. How would a one-tailed test affect the critical value?
4. What implications would failing to reject the null hypothesis have for this study?
5. What alternative methods could be used if the assumptions for the \( t \)-test are violated?

### Tip:
Always check if the assumptions for a \( t \)-test (such as normality and equal variances) are satisfied, as they can influence the validity of the results.

An observational study of Alzheimer's disease (AD) obtained data from 15 AD patients exhibiting moderate dementia and selected a group of 10 control individuals without AD. AD is a progressive neurodegenerative disease of the elderly and advancing age is known to be a primary risk factor in AD diagnosis. Therefore, it was crucial for the study's credibility to examine whether the ages in the AD group might be significantly different than in the control group. The ages of the subjects in years are summarized in the R Output below.

Variable	n	Mean	sd	Minimum	Q1	Median	Q3	Maximum
Alzheimers	15	74.29	6.62	77.00	79.25	87.00	92.25	93.00
Control	10	65.37	12.94	54.00	56.00	65.00	82.00	89.00

We want to test if the average age in the Alzheimer's group is significantly different than the control group.

(b) Find the value of the test statistic. in 3 decimals
(c) Find the 5% critical value. in 3 decimals

Learn to perform an independent samples t-test to determine if there's a significant difference in average age between Alzheimer's and control groups. This problem includes hypothesis testing, pooled standard deviation calculation, and critical value interpretation, making it ideal for undergraduate statistics students.

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