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To test the claim that there is a significant difference in IQs between spouses, we can conduct a paired \( t \)-test using the provided data. Here’s how we approach this problem:

### Step-by-Step Solution

1. **Define the Hypotheses**:
   - Null Hypothesis (\( H_0 \)): There is no significant difference in IQs between spouses (\( \mu_d = 0 \)), where \( \mu_d \) is the mean of the differences between IQs of each pair.
   - Alternative Hypothesis (\( H_a \)): There is a significant difference in IQs between spouses (\( \mu_d \neq 0 \)).

2. **Calculate the Differences and the Mean of Differences**:
   For each pair of spouses, we find the difference:
   \[
   d_i = \text{IQ of Spouse 1} - \text{IQ of Spouse 2}
   \]
   and then calculate the mean \( \bar{d} \) and standard deviation \( s_d \) of these differences.

3. **Compute the Test Statistic**:
   The test statistic for a paired \( t \)-test is calculated as:
   \[
   t = \frac{\bar{d}}{s_d / \sqrt{n}}
   \]
   where:
   - \( \bar{d} \) is the mean of the differences,
   - \( s_d \) is the standard deviation of the differences,
   - \( n \) is the number of pairs (in this case, \( n = 9 \)).

4. **Compare with Critical Value**:
   Since the level of significance is \( \alpha = 0.05 \), we check the calculated \( t \)-statistic against the critical value for a two-tailed \( t \)-test with \( n-1 = 8 \) degrees of freedom.

Let's calculate each step.The calculated \( t \)-statistic is approximately \( 2.774 \), and the critical \( t \)-value for a two-tailed test at a 0.05 significance level with 8 degrees of freedom is approximately \( 2.306 \).

### Conclusion
Since \( t = 2.774 \) is greater than the critical value of \( 2.306 \), we reject the null hypothesis. This indicates that there is a statistically significant difference in IQs between spouses at the 0.05 significance level.

Would you like further details or have any questions?

Here are some related questions to explore:
1. How is the paired \( t \)-test different from an independent \( t \)-test?
2. What assumptions are made in a paired \( t \)-test?
3. How would the test conclusion change if we used a one-tailed test?
4. What is the effect size, and how is it calculated for paired tests?
5. How does sample size affect the power of the test?

**Tip:** Always check if data is normally distributed when using the \( t \)-test, especially with small sample sizes.

A psychology graduate student wants to test the claim that there is a significant difference between the IQs of spouses using a standard IQ test. The IQs of 9 married couples are measured, and a paired t-test is used to evaluate the claim with a 0.05 significance level. Compute the value of the test statistic and determine if there is a significant difference.

Perform a paired t-test to determine if there is a significant difference in IQs between spouses based on data from 9 married couples. This analysis covers hypothesis testing, test statistic calculation, and comparison with the critical value at a 0.05 significance level.

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