Let's solve these problems step by step.

We are given the following information:

• The mean mortgage debt, $\mu = 185,000$
• The standard deviation of mortgage debt, $\sigma = 85,000$

We will use the standard normal distribution (Z-scores) to solve the problems.

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(a) What is the probability of randomly selecting a Canadian with mortgage debt that exceeds $243,600$?

To find this, we will:

1. Convert the debt amount into a Z-score using the formula: $Z = \frac{X - \mu}{\sigma}$ where $X$ is the given debt amount, $\mu$ is the mean, and $\sigma$ is the standard deviation.

2. Use the standard normal distribution table to find the corresponding probability.

For $X = 243,600$: $Z = \frac{243,600 - 185,000}{85,000} = \frac{58,600}{85,000} \approx 0.6894$

Using the Z-table (or a calculator), the cumulative probability for $Z = 0.6894$ is approximately $P(Z \leq 0.6894) = 0.7547$.

Therefore, the probability of exceeding this debt amount is: $P(Z > 0.6894) = 1 - 0.7547 = 0.2453$

Thus, the probability is approximately 0.2453.

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(b) What is the probability of randomly selecting a Canadian with mortgage debt below $97,000$?

Again, we will:

1. Calculate the Z-score for $X = 97,000$: $Z = \frac{97,000 - 185,000}{85,000} = \frac{-88,000}{85,000} \approx -1.0353$

2. Use the Z-table to find the cumulative probability for $Z = -1.0353$.

The cumulative probability for $Z = -1.0353$ is approximately $P(Z \leq -1.0353) = 0.1506$.

Thus, the probability of selecting a Canadian with a mortgage debt below $97,000$ is approximately 0.1506.

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(c) Determine the minimum mortgage debt owing by the 24% of Canadians with the largest mortgages.

To find this:

1. We need to find the Z-score corresponding to the top 24% (i.e., the 76th percentile). From the Z-table, the Z-score for $P(Z \leq 0.76) = 0.76$ is approximately 0.7063.

2. Convert this Z-score back into the mortgage debt amount using the formula: $X = \mu + Z \cdot \sigma$ where $\mu = 185,000$ and $\sigma = 85,000$.

$X = 185,000 + 0.7063 \cdot 85,000 = 185,000 + 60,035.5 \approx 245,036$

Thus, the minimum mortgage debt owing by the top 24% of Canadians is approximately $245,036.

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Summary of answers:

• (a) Probability of mortgage debt exceeding $243,600: 0.2453
• (b) Probability of mortgage debt below $97,000: 0.1506
• (c) Minimum mortgage debt for the top 24%: $245,036

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Would you like further explanation or clarification on any of these steps?

Here are 5 related questions to further explore:

1. How does increasing the standard deviation affect the probabilities in parts (a) and (b)?
2. What is the probability of selecting a Canadian with mortgage debt between $97,000 and $243,600?
3. What mortgage debt amount represents the 90th percentile of Canadians?
4. If Albertans had a standard deviation of $90,000 instead, how would that affect the probability in part (a)?
5. How would the minimum mortgage debt for the top 24% change if the average mortgage debt were $200,000?

Tip: When working with normal distributions, always convert the values into Z-scores to use the standard normal distribution table.