The average monthly mortgage payment including principal and interest is $982 in the United States. If the standard deviation is approximately $180 and the mortgage payments are approximately normally distributed, find the probability that a randomly selected monthly payment is:

a. More than $1000
b. More than $1475
c. Between $800 and $1150

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To solve this problem, we can use the standard normal distribution, Z-score formula, and percentile table.

Step 1: Calculate the Z-score for each value using the formula:
Z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.

Given:
Mean, μ = $982
Standard deviation, σ = $180

a. More than $1000
Z = (X - μ) / σ = (1000 - 982) / 180 = 0.1

b. More than $1475
Z = (X - μ) / σ = (1475 - 982) / 180 = 2.72

c. Between $800 and $1150
Z1 = (X1 - μ) / σ = (800 - 982) / 180 = -1.01
Z2 = (X2 - μ) / σ = (1150 - 982) / 180 = 3.76

Step 2: Use the percentile table or a calculator to find the probabilities associated with the Z-scores obtained.

a. To find the probability that a randomly selected monthly payment is more than $1000, we need to calculate the probability of Z > 0.1. From the Z-table, the value for Z = 0.1 is approximately 0.5398. However, we need the probability of Z > 0.1, so we subtract this value from 1.
P(Z > 0.1) = 1 - 0.5398 = 0.4602

b. To find the probability that a randomly selected monthly payment is more than $1475, we need to calculate the probability of Z > 2.72. From the Z-table, the value for Z = 2.72 is approximately 0.9969. However, we need the probability of Z > 2.72, so we subtract this value from 1.
P(Z > 2.72) = 1 - 0.9969 = 0.0031

c. To find the probability that a randomly selected monthly payment is between $800 and $1150, we need to calculate the probability of Z1 < Z < Z2. Using the Z-table, we find that the probability of Z < -1.01 is approximately 0.1562, and the probability of Z < 3.76 is approximately 0.9999. Therefore, to find the probability between these two values, we subtract the smaller probability from the larger one.
P(Z1 < Z < Z2) = 0.9999 - 0.1562 = 0.8437

Therefore:
a. The probability that a randomly selected monthly payment is more than $1000 is approximately 0.4602, or 46.02%.
b. The probability that a randomly selected monthly payment is more than $1475 is approximately 0.0031, or 0.31%.
c. The probability that a randomly selected monthly payment is between $800 and $1150 is approximately 0.8437, or 84.37%.

To find the probabilities in this situation, we can use the concept of z-scores. A z-score tells us how many standard deviations a particular value is from the mean. We can then use a z-table or statistical software to find the corresponding probabilities.

1. To find the probability of a randomly selected monthly payment being more than $1000, we need to first calculate the z-score for this value. We can use the formula: z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation.

In this case, x = $1000, μ = $982, and σ = $180. Plugging these values into the formula, we get:

z = ($1000 - $982) / $180 = 0.1

Now, we can look up the probability associated with a z-score of 0.1 in a z-table. The z-table provides the area to the left of the z-score, so we subtract the probability from 1 to get the probability of being more than $1000. Alternatively, we can use statistical software to find this probability.

2. Similarly, to find the probability of a payment being more than $1475, we calculate the z-score:

z = ($1475 - $982) / $180 = 2.75

Again, we look up the probability associated with a z-score of 2.75 (or more accurately, the probability associated with a z-score greater than 2.75) in a z-table or use statistical software.

3. To find the probability of a payment being between $800 and $1150, we need to calculate the z-score for both values:

For $800:
z1 = ($800 - $982) / $180 = -1.01

For $1150:
z2 = ($1150 - $982) / $180 = 1.88

Now, we need to find the probability of being between these two z-scores. Since the standard normal distribution is symmetric, we can find the area to the left of each z-score and then subtract the smaller probability from the larger probability.

Using a z-table or statistical software, we find the probability associated with a z-score of -1.01 and a z-score of 1.88, and then subtract the smaller probability from the larger one to get the probability of the range.

Note: In practice, it's often more efficient and accurate to use statistical software or calculators that can directly provide the desired probabilities, but understanding the concept of z-scores and using a z-table is still valuable.