the average monthly mortgage payment for all homeowners in a city is 2760. suppose that the distribution of monthly mortgages paid by homeowners in this city follow an approximately normal distribution with a mean of 2760 and a standard deviation of 460. find to 4 decimal places that the probability that the monthly mortgage paid by a randomly selected homeowner from this city is less than 1300
To find the probability that the monthly mortgage paid by a randomly selected homeowner from this city is less than 1300, we will use the z-score formula and the standard normal distribution.
First, we need to calculate the z-score using the formula:
z = (x - μ) / σ
Where:
x = 1300 (the value we want to find the probability for)
μ = 2760 (mean monthly mortgage payment)
σ = 460 (standard deviation)
Plugging in the values:
z = (1300 - 2760) / 460
z = -1.8696 (rounded to 4 decimal places)
Now, we will use a standard normal distribution table or a calculator to find the corresponding probability for this z-score.
Looking up the z-score of -1.8696 in the standard normal distribution table, we find that the corresponding probability is 0.0307. Therefore, the probability that the monthly mortgage paid by a randomly selected homeowner from this city is less than 1300 is approximately 0.0307.
To find the probability that the monthly mortgage paid by a randomly selected homeowner from this city is less than $1300, we need to standardize the value using the z-score formula.
The formula for the z-score is:
z = (x - μ) / σ
Where:
x = the value we want to standardize (1300 in this case)
μ = the mean (2760 in this case)
σ = the standard deviation (460 in this case)
Using these values, we can calculate the z-score:
z = (1300 - 2760) / 460
z = -1.8261
To find the probability associated with this z-score, we can reference the z-table or use a calculator that can compute normal distribution probabilities.
Using a calculator or a z-table, we can find that the probability associated with a z-score of -1.8261 is approximately 0.0339.
Therefore, the probability that the monthly mortgage paid by a randomly selected homeowner from this city is less than $1300 is approximately 0.0339, or 3.39%.
To find the probability that the monthly mortgage paid by a randomly selected homeowner from this city is less than 1300, we need to calculate the z-score and then find the corresponding area under the standard normal distribution curve.
Step 1: Calculate the z-score
The z-score is calculated using the formula:
z = (X - μ) / σ
where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
In this case, X = 1300, μ = 2760, and σ = 460.
z = (1300 - 2760) / 460
z = -1.826
Step 2: Find the area under the standard normal distribution curve
We need to find the area to the left of the z-score we calculated. This represents the probability that a randomly selected homeowner has a monthly mortgage payment less than 1300.
We can use a standard normal distribution table or a calculator to find the corresponding area.
Using a standard normal distribution table or a calculator, we find that the area to the left of the z-score -1.826 is approximately 0.0341.
Therefore, the probability that the monthly mortgage paid by a randomly selected homeowner from this city is less than 1300 is approximately 0.0341, or 3.41% (rounded to 4 decimal places).