- Calculate \(z\)-scores using \(\mu=982\) and \(\sigma=180\).
- For part (a), \(P(X > 1220) = P(Z > 1.32) \approx \boxed{0.0934}\).
- For part (b), \(P(610 < X < 1050) = P(-2.07 < Z < 0.38) = P(Z < 0.38) - P(Z < -2.07) \approx \boxed{0.6288}\).
Explanation
- Problem analysis and methodology
We analyze a normal distribution where monthly mortgage payments \(X\) follow a distribution with a mean \(\mu = 982\) and a standard deviation \(\sigma = 180\). To find specific probabilities for the normal distribution, we first transform our values into standard normal \(z\)-scores using the formula:
\[z = \frac{X - \mu}{\sigma}\]
Once the \(z\)-scores are calculated and rounded to two decimal places, we can determine the corresponding probabilities using the cumulative distribution function (CDF) of the standard normal distribution.
- Calculating probability for (a): More than \$1220
To find the probability that the monthly payment is more than \(\$1220\), we calculate the \(z\)-score for \(X = 1220\):
\[z = \frac{1220 - 982}{180} = \frac{238}{180} \approx 1.3222\]
Rounding to two decimal places, we get \(z = 1.32\).
Now, we find \(P(X > 1220) = P(Z > 1.32)\). Using the standard normal distribution, the probability \(P(Z < 1.32)\) is approximately \(0.9066\).
Thus, \(P(Z > 1.32) = 1 - 0.9066 = 0.0934\).
- Calculating probability for (b): Between \$610 and \$1050
To find the probability that the monthly payment is between \(\$610\) and \(\$1050\), we calculate the \(z\)-scores for both boundaries:
For \(X = 610\):
\[z_1 = \frac{610 - 982}{180} = \frac{-372}{180} \approx -2.0667 \approx -2.07\]
For \(X = 1050\):
\[z_2 = \frac{1050 - 982}{180} = \frac{68}{180} \approx 0.3778 \approx 0.38\]
We now find \(P(-2.07 < Z < 0.38) = P(Z < 0.38) - P(Z < -2.07)\).
Using standard normal distribution tables, \(P(Z < 0.38) \approx 0.6480\) and \(P(Z < -2.07) \approx 0.0192\).
Subtracting these values:
\[0.6480 - 0.0192 = 0.6288\]
Thus, the probability of the payment being between \(\$610\) and \(\$1050\) is \(0.6288\).
Examples
Understanding the probability distribution of financial data like mortgage payments is essential for banks and insurance companies to manage risk. For instance, by calculating the proportion of customers falling into certain payment brackets, a lender can estimate how many borrowers might be affected by changes in interest rates or predict the likelihood of default, helping them to structure their loan offerings more effectively and maintain financial stability.