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To solve part (a) of the question, we need to test the hypothesis that the variances of the sale prices in Dublin 1 and Dublin 3 are equal, using the provided sample data.

### Hypotheses:
- Null hypothesis (\(H_0\)): \(\sigma_1^2 = \sigma_3^2\) (the population variances of Dublin 1 and Dublin 3 are equal).
- Alternative hypothesis (\(H_a\)): \(\sigma_1^2 \neq \sigma_3^2\) (the population variances of Dublin 1 and Dublin 3 are not equal).

### Test statistic:
Since we are comparing two variances, we use an F-test for equality of variances. The F-statistic is computed as the ratio of the sample variances:

\[
F = \frac{s_1^2}{s_3^2}
\]

where:
- \( s_1^2 \) is the sample variance of Dublin 1 (calculated from the standard deviation provided, which is \( 56.03 \)).
- \( s_3^2 \) is the sample variance of Dublin 3 (calculated from the standard deviation provided, which is \( 49.18 \)).

### Step 1: Calculate the variances.
\[
s_1^2 = (56.03)^2
\]
\[
s_3^2 = (49.18)^2
\]

### Step 2: Calculate the F-statistic.
\[
F = \frac{s_1^2}{s_3^2}
\]

### Step 3: Compare with critical value.
Given that the degrees of freedom for the numerator and denominator are both \( n - 1 = 22 - 1 = 21 \), and a significance level \(\alpha = 0.05\), the critical value for the F-distribution is provided as \( F_{critical} = 2.08418862 \).

Let me first calculate the variances and then determine the F-statistic.The calculated F-statistic is approximately \( F = 1.298 \).

### Step 4: Decision
We compare this F-statistic with the critical value \( F_{critical} = 2.084 \).

Since the F-statistic (1.298) is less than the critical value (2.084), we **fail to reject the null hypothesis**. This means there is not enough evidence to conclude that the variances of the two-bedroom apartment prices in Dublin 1 and Dublin 3 are significantly different.

### Assumptions for the F-test:
1. The data in each sample (Dublin 1 and Dublin 3) must be independent of each other.
2. The samples must be drawn from normally distributed populations.
3. The data should not have any extreme outliers.

Would you like to explore further details, or do you have any questions?

Here are 5 follow-up questions you might want to explore:
1. How do we test if the means of two populations are significantly different?
2. How can we check if the normality assumption holds for this data?
3. What happens if the variances are found to be significantly different?
4. Can a different statistical test be used if the normality assumption is violated?
5. How does the sample size affect the F-test?

**Tip**: The F-test is sensitive to violations of normality, so checking for normal distribution or using a robust test might be necessary.

Test the hypothesis that the population variance of the two-bedroom apartment sale price in Dublin 1 is equal to the population variance in Dublin 3.

This problem uses the F-test to examine whether the variances of two-bedroom apartment sale prices in Dublin 1 and Dublin 3 are significantly different. Given 22 samples from each region, we calculate the F-statistic using sample standard deviations and compare it to the critical F-value.

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