Final answer:
To address the mortgage-related probabilities, z-scores are calculated based on the given mean and standard deviation followed by referencing the standard normal distribution table; to find the amount owed by the top 23%, the z-score for the 77th percentile is utilized to derive the minimum mortgage debt.
Step-by-step explanation:
To find the probability of randomly selecting a Canadian with a mortgage debt that exceeds $246,400, we need to calculate the z-score and then use a standard normal distribution table.
Calculation for Part a:
Z= (X - μ) / σ
Where Z is the z-score, X is the value of interest ($246,400), μ is the mean average debt ($184,000), and σ is the standard deviation ($92,000).
Z= ($246,400 - $184,000) / $92,000
Z= $62,400 / $92,000
Z= 0.6783
Looking up the z-score of 0.6783 in the standard normal distribution table, we find the area to the left of Z is approximately 0.7517. Hence, the probability of a Canadian owing more than $246,400 is 1 - 0.7517 = 0.2483 or 24.83%.
Calculation for Part b:
Similarly, to find the probability of having a debt below $94,000, we calculate the z-score for $94,000.
Z= ($94,000 - $184,000) / $92,000
Z= -$90,000 / $92,000
Z= -0.9783
Using the standard normal distribution table, the area to the left of Z is approximately 0.1635. Therefore, the probability of a Canadian having debt below $94,000 is 16.35%.
Calculation for Part c:
To find the minimum mortgage debt for the top 23% of Canadians, we look for the z-score that corresponds to the 77th percentile (100% - 23%) in the standard normal distribution table, which is approximately 0.74. We then translate the z-score back into the mortgage debt value.
Z = 0.74
Mortgage Debt = Z * σ + μ
Mortgage Debt = 0.74 * $92,000 + $184,000
Mortgage Debt = $68,080 + $184,000
Mortgage Debt = $252,080
The minimum mortgage debt for the 23% of Canadians with the largest mortgages is approximately $252,080.