Questions: Monthly Mortgage Payments 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 probabilities. Use a TI-83 Plus/TI-84 Plus calculator and round the answers to at least four decimal places. Part 1 of 2 (a) The selected monthly payment is more than 1400. P(X>1400)=.0101 Alternate Answer: P(X>1400)=0.0101 Part 2 of 2 (b) The selected monthly payment is between 650 and 975. P(650<X<975)=

Monthly Mortgage Payments 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 probabilities. Use a TI-83 Plus/TI-84 Plus calculator and round the answers to at least four decimal places.

Part 1 of 2
(a) The selected monthly payment is more than 1400.
P(X>1400)=.0101

Alternate Answer:
P(X>1400)=0.0101

Part 2 of 2
(b) The selected monthly payment is between 650 and 975.
P(650<X<975)=
Transcript text: Monthly Mortgage Payments 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 probabilities. Use a TI-83 Plus/TI-84 Plus calculator and round the answers to at least four decimal places. Part 1 of 2 (a) The selected monthly payment is more than $1400. \[ P(X>1400)=.0101 \] Alternate Answer: \[ P(X>1400)=0.0101 \] Part 2 of 2 (b) The selected monthly payment is between $650 and $975. \[ P(650
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Solution

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Solution Steps

Step 1: Calculate P(X>1400) P(X > 1400)

To find the probability that the selected monthly payment is greater than \$1400, we first calculate the Z-score for \( X = 1400 \):

Z=Xμσ=14009821802.3222 Z = \frac{X - \mu}{\sigma} = \frac{1400 - 982}{180} \approx 2.3222

Using the standard normal distribution, we find:

P(X>1400)=1P(Z<2.3222)10.9899=0.0101 P(X > 1400) = 1 - P(Z < 2.3222) \approx 1 - 0.9899 = 0.0101

Step 2: Calculate P(650<X<975) P(650 < X < 975)

Next, we calculate the probability that the selected monthly payment is between \$650 and \$975. We find the Z-scores for both values:

For X=650 X = 650 :

Z650=6509821801.8444 Z_{650} = \frac{650 - 982}{180} \approx -1.8444

For X=975 X = 975 :

Z975=9759821800.0389 Z_{975} = \frac{975 - 982}{180} \approx -0.0389

Now, we can find the probability:

P(650<X<975)=P(Z<0.0389)P(Z<1.8444)0.48410.0322=0.4519 P(650 < X < 975) = P(Z < -0.0389) - P(Z < -1.8444) \approx 0.4841 - 0.0322 = 0.4519

Final Answer

  • For part (a): P(X>1400)0.0101 P(X > 1400) \approx 0.0101
  • For part (b): P(650<X<975)0.4519 P(650 < X < 975) \approx 0.4519

Thus, the final answers are:

P(X>1400)0.0101 \boxed{P(X > 1400) \approx 0.0101} P(650<X<975)0.4519 \boxed{P(650 < X < 975) \approx 0.4519}

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