Question 1179741
Here's how to solve this problem, including calculating the confidence interval and conducting the hypothesis test:

**1. Given Information:**

* **Exeter:**
    * n1 = 21
    * x̄1 = £116,900
    * s1 = £2,300
* **Cardiff:**
    * n2 = 26
    * x̄2 = £114,000
    * s2 = £1,750
* Confidence level = 90%
* Significance level (α) = 0.10
* Assume normal distribution and equal variances.

**2. Calculate the Pooled Standard Deviation (sp):**

Since we assume equal variances, we use the pooled standard deviation:

sp = √[((n1 - 1) * s1²) + ((n2 - 1) * s2²)) / (n1 + n2 - 2)]

sp = √[((20 * 2300²) + (25 * 1750²)) / (21 + 26 - 2)]
sp = √[(105800000 + 76562500) / 45]
sp = √[182362500 / 45]
sp = √4052500
sp ≈ £2,013.08

**3. Calculate the Standard Error of the Difference (SE):**

SE = sp * √(1/n1 + 1/n2)
SE = 2013.08 * √(1/21 + 1/26)
SE = 2013.08 * √(0.0476 + 0.0385)
SE = 2013.08 * √0.0861
SE = 2013.08 * 0.2934
SE ≈ £590.69

**4. Find the Critical t-Value:**

* Degrees of freedom (df) = n1 + n2 - 2 = 21 + 26 - 2 = 45
* For a 90% confidence interval (α = 0.10, two-tailed), the critical t-value (t*) for df = 45 is approximately 1.68.

**5. Calculate the Margin of Error (E):**

E = t* * SE
E = 1.68 * 590.69
E ≈ £992.36

**6. Construct the Confidence Interval:**

* Difference in sample means (x̄1 - x̄2) = 116,900 - 114,000 = £2,900
* Lower Bound = (x̄1 - x̄2) - E = 2,900 - 992.36 = £1,907.64
* Upper Bound = (x̄1 - x̄2) + E = 2,900 + 992.36 = £3,892.36

**90% Confidence Interval: (£1,907.64, £3,892.36)**

**7. Hypothesis Test:**

* **Null Hypothesis (H0):** μ1 - μ2 = 0 (There is no difference in mean prices)
* **Alternative Hypothesis (H1):** μ1 - μ2 ≠ 0 (There is a difference in mean prices)
* Significance level (α) = 0.10
* Test statistic (t): t = (x̄1 - x̄2) / SE

t = 2900 / 590.69
t ≈ 4.91

**8. Find the Critical t-Values:**

* df = 45
* For α = 0.10 (two-tailed), the critical t-values are approximately ±1.68.

**9. Make a Decision:**

* Calculated t-value (4.91) > critical t-value (1.68).
* Therefore, we reject the null hypothesis.

**10. Conclusion:**

There is sufficient evidence at the α = 0.10 level to conclude that there is a difference in the mean prices of mid-range homes between Exeter and Cardiff.

**Answers:**

* 90% Confidence Interval: (£1,907.64, £3,892.36)
* Hypothesis Test: Reject the null hypothesis. There is a statistically significant difference.