Problem 6 The average monthly mortgage pay... [FREE SOLUTION]

Chapter 6: Problem 6

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 $\$ 1150$

Short Answer

Expert verified

a. 45.98%, b. 0.31%, c. 66.76%

Step by step solution

Calculate Z-score for Part a

We need to find the Z-score for a monthly payment of $ \\(1000 $. The Z-score is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \] where $ X $ is the value, $ \mu $ is the mean, and $ \sigma $ is the standard deviation.For $ X = \$1000 \), $ \mu = \\(982 $, and $ \sigma = \$180 \):\[ Z = \frac{1000 - 982}{180} \approx 0.10 \]

Find Probability for Part a

With the Z-score calculated, we now look up $ Z = 0.10 $ in a standard normal distribution table or use a calculator to find the probability. The probability $ P(Z > 0.10) $ is approximately $ 0.4602 $, since tables typically provide $ P(Z < 0.10) \approx 0.5402 $.Thus, $ P(X > 1000) = 1 - 0.5402 = 0.4598 $.

Calculate Z-score for Part b

Now, calculate the Z-score for a monthly payment of $ \$1475 $:\[ Z = \frac{1475 - 982}{180} \approx 2.74 \]

Find Probability for Part b

For $ Z = 2.74 $, the probability $ P(Z < 2.74) \approx 0.9969 $ using a standard normal distribution.Hence, $ P(X > 1475) = 1 - 0.9969 = 0.0031 $.

Calculate Z-scores for Part c

For this range, we need two Z-scores. For $ X = 800 $:\[ Z = \frac{800 - 982}{180} \approx -1.01 \]For $ X = 1150 $:\[ Z = \frac{1150 - 982}{180} \approx 0.93 \]

Find Probability for Part c

Look up the probabilities for these Z-scores. For $ Z = -1.01 $, $ P(Z < -1.01) \approx 0.1562 $.For $ Z = 0.93 $, $ P(Z < 0.93) \approx 0.8238 $.Then, $ P(800 < X < 1150) = P(Z < 0.93) - P(Z < -1.01) = 0.8238 - 0.1562 = 0.6676 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-score

The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. To determine a Z-score, the formula \[ Z = \frac{X - \mu}{\sigma} \]is used, where:

$ X $ is the value being analyzed
$ \mu $ is the mean of the dataset
$ \sigma $ is the standard deviation of the dataset

Z-scores indicate how many standard deviations a value lies from the mean. For example, in the mortgage exercise:

A Z-score of $ 0.10 $ suggests that the payment of $ \\(1000 $ is just above the average $ \$982 \).
A higher Z-score of $ 2.74 $ for $ \\(1475 $ indicates it's quite a bit higher than the average.
Z-scores can be negative, which means the value is below the average. This is shown by a score of $ -1.01 $ for $ \$800 \).

Standard Deviation

Standard deviation is crucial in statistics as it shows how much variation there is from the average. In simpler terms, it tells us how spread out numbers are in a dataset. The standard deviation ($ \sigma $) used here is $ \\(180 $.
In the context of mortgage payments:

If most monthly payments cluster around the mean of $ \$982 \), the standard deviation will help determine how typical or atypical a payment value is.
A low standard deviation would imply that most payments are close to the average, while a higher standard deviation indicates that there is a wider spread of payment amounts.
In this example, the standard deviation helps us compute Z-scores, which are essential in determining how unusual certain payments, like $ \\(1000 $ or $ \$1475 \), are.

Probability

Probability helps us determine the likelihood of a particular outcome. When we analyze normally distributed data, we frequently use Z-scores to find probabilities. In our example, we calculated probabilities for three scenarios:

The chance that a payment is more than $ \\(1000 $ is approximately $ 0.4598 $. This means there's almost a $ 46% $ chance a random payment is over $ \$1000 \).
For payments exceeding $ \\(1475 $, the probability is much smaller at $ 0.0031 $, suggesting such high payments are rare.
Between $ \$800 \) and $ \$1150 $, the likelihood increases significantly to $ 0.6676 $, indicating that typical payments fall within this range with a probability of more than $ 66% $.

Recognizing these probabilities helps us understand how likely different payment values are within the given distribution.

Statistics

Statistics is a branch of mathematics focusing on data collection, analysis, interpretation, and presentation. In the context of our example involving mortgage payments, statistics provides tools to understand data patterns.
Here are key takeaways regarding this topic:

Statistical measures such as the mean and standard deviation allow us to summarize data comprehensively.
The normal distribution, often depicted as a bell curve, is vital in analyzing data that clusters around a mean鈥攕hown by our mortgage payments.
Z-scores and probabilities aid in exploring and predicting data behaviors, making statistics incredibly powerful for real-world applications like predicting financial data trends.

Overall, statistics turns raw numbers into meaningful insights, enabling informed decisions based on data.

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Short Answer

Step by step solution

Calculate Z-score for Part a

Find Probability for Part a

Calculate Z-score for Part b

Find Probability for Part b

Calculate Z-scores for Part c

Find Probability for Part c

Key Concepts

Z-score

Standard Deviation

Probability

Statistics

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