Question

Suppose Canadian home-owners owe an average of $195,000 on their mortgages. Assume that mortgage debt is normally distributed in Canada with a standard deviation of $98,000.

Standard Normal Distribution Table

a. Albertans are reported to owe $245,800 in mortgage debt, much higher than the Canadian average. What is the probability of randomly selecting a Canadian with mortgage debt that exceeds $245,800?

Round to four decimal places if necessary

b. What is the probability of randomly selecting a Canadian with mortgage debt below $96,000?

Round to four decimal places if necessary

c. Determine the minimum mortgage debt owing by the 18% of Canadians with the largest mortgages.

Round to the nearest dollar

Answer 1

a. The probability of randomly selecting a Canadian with mortgage debt that exceeds $245,800 is approximately 0.6997. b. The probability of randomly selecting a Canadian with mortgage debt below $96,000 is approximately 0.1587. c. The minimum mortgage debt owing by the 18% of Canadians with the largest mortgages is approximately $281,198.

Step-by-step explanation:

a. To find the probability of randomly selecting a Canadian with mortgage debt that exceeds $245,800, we need to calculate the z-score using the formula: z = (x - μ) / σ, where x is the value we want to find the probability for, μ is the average mortgage debt, and σ is the standard deviation. Plugging in the values: z = (245800 - 195000) / 98000 = 0.52. Using a standard normal distribution table, the probability corresponding to a z-score of 0.52 is approximately 0.6997. So, the probability of randomly selecting a Canadian with mortgage debt that exceeds $245,800 is approximately 0.6997.

b. To find the probability of randomly selecting a Canadian with mortgage debt below $96,000, we can follow the same steps as in part a. The z-score is calculated as: z = (96000 - 195000) / 98000 = -1.00. Using the standard normal distribution table, the probability corresponding to a z-score of -1.00 is approximately 0.1587. So, the probability of randomly selecting a Canadian with mortgage debt below $96,000 is approximately 0.1587.

c. To determine the minimum mortgage debt owing by the 18% of Canadians with the largest mortgages, we need to find the z-score corresponding to the 82nd percentile (100% - 18% = 82%). Using the standard normal distribution table, the z-score corresponding to the 82nd percentile is approximately 0.926. We can then use the z-score formula to find the minimum mortgage debt: x = μ + z * σ, where x is the minimum mortgage debt, μ is the average mortgage debt, z is the z-score, and σ is the standard deviation. Plugging in the values: x = 195000 + 0.926 * 98000 = 281198. The minimum mortgage debt owing by the 18% of Canadians with the largest mortgages is approximately $281,198.