To buy a home, people usually take out loans called mortgages - Leaving Cert Mathematics - Question 7 - 2012

The interest rates being paid have reflected a shift as well. It appears that in 2009, interest rates ranged primarily between 2.3% and 4.1%, while by 2011, they have shifted upwards, mostly being between 4% and 6%. This suggests a potential link with the increase in borrowing difficulties.

The relationship between the arrears rate and the interest rate indicates a stronger correlation in 2011 compared to 2009. This suggests that as interest rates increase, the number of mortgages in arrears also rises, highlighting a potential impact of financial strain on borrowers.

To accurately estimate the median interest rate for mortgage holders in September 2011, it would be crucial to know how many mortgage holders are represented by each point on the relevant diagrams. Understanding the distribution of borrowers along the interest rate spectrum would provide insight into the central tendency.

In this context, "direction of causality" refers to whether higher arrears rates cause interest rates to rise, or if increasing interest rates lead to a higher number of arrears. Establishing this direction is essential for understanding how these two variables influence each other.

The probability that a randomly selected property will be in negative equity can be calculated as:

$P(Neg.E) = \frac{145414}{475136} = 0.306 \text{ (to two decimal places)}$

The probability that a property taken from those in negative equity is also in arrears is:

$P(Arrears | Neg.E) = \frac{11644}{145414} = 0.08 \text{ (to two decimal places)}$

To find this probability, we calculate it as:

$P(Neg.E | Arrears) = \frac{11644}{24011} = 0.48 \text{ (to two decimal places)}$

Null Hypothesis (H0): Proportion in negative equity unchanged: $p = 0.031$

Alternative Hypothesis (H1): Proportion has changed: $p \neq 0.031$

Conducting a test based on a sample of 2000 properties:

1. Calculate the observed proportion: $ext{Observed proportion} = \frac{552}{2000} = 0.276$
2. Calculate the 95% margin of error: $0.031 \pm 1.96 \sqrt{\frac{0.031(1 - 0.031)}{2000}} = 0.031 \pm 0.0224$
3. Compare observed proportion with the margin of error: If $0.276$ lies outside of $[0.0086, 0.0534]$, we reject the null hypothesis.

Conclusion: Since $0.276$ falls outside the margin of error, reject H0: the proportion in negative equity has changed.