The XO Group Inc. conducted a 2015 survey of 13000 brides and grooms married in the United States and found that the average cost of a wedding is $29858 (XO Group website). Assume that the cost of a wedding is normally distributed with a mean of 29858  and a standard deviation of 5600. 

a.  What is the probability that a wedding costs less than $20000 (to 4 decimals)? 

b.  What is the probability that a wedding costs between 20000  and 30000 (to 4 decimals)? 

c.  For a wedding to be among the 5% most expensive, how much would it have to cost (to the nearest whole number)?

Given the problem, we know the cost of a wedding is normally distributed with:

- Mean $ \mu = 29858 $
- Standard deviation $ \sigma = 5600 $

We will use the Z-score formula to standardize the given values for each part of the question.

The **Z-score formula** is:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- $ X $ is the value of interest
- $ \mu $ is the mean
- $ \sigma $ is the standard deviation

### (a) Probability that a wedding costs less than \$20,000:

1. Calculate the Z-score for $ X = 20000 $:

\[
Z = \frac{20000 - 29858}{5600} = \frac{-9858}{5600} \approx -1.7604
\]

2. Using the standard normal distribution table (or a calculator), the cumulative probability corresponding to $ Z = -1.7604 $ is approximately 0.0392.

Thus, the probability that a wedding costs less than \$20,000 is **0.0392** (or 3.92%).

---

### (b) Probability that a wedding costs between \$20,000 and \$30,000:

1. Calculate the Z-scores for $ X = 20000 $ and $ X = 30000 $:

   For $ X = 20000 $, we already found that $ Z \approx -1.7604 $.

   For $ X = 30000 $:

\[
Z = \frac{30000 - 29858}{5600} = \frac{142}{5600} \approx 0.0254
\]

2. Using the standard normal distribution table:
   - The cumulative probability for $ Z = -1.7604 $ is 0.0392.
   - The cumulative probability for $ Z = 0.0254 $ is approximately 0.5101.

3. The probability that a wedding costs between \$20,000 and \$30,000 is:

\[
P(20000 < X < 30000) = P(Z = 0.0254) - P(Z = -1.7604) = 0.5101 - 0.0392 = 0.4709
\]

Thus, the probability that a wedding costs between \$20,000 and \$30,000 is **0.4709** (or 47.09%).

---

### (c) Cost for a wedding to be among the 5% most expensive:

1. To be among the 5% most expensive weddings, we need to find the cost corresponding to the 95th percentile of the distribution. Using a standard normal distribution table, the Z-score for the 95th percentile is approximately **1.645**.

2. Use the Z-score formula to find the corresponding value of $ X $:

\[
1.645 = \frac{X - 29858}{5600}
\]

3. Solve for $ X $:

\[
X = 1.645 \times 5600 + 29858 \approx 9202 + 29858 = 39060
\]

Thus, a wedding would need to cost **\$39,060** or more to be among the 5% most expensive.

---

### Summary:
- **(a)** Probability that a wedding costs less than \$20,000: **0.0392**
- **(b)** Probability that a wedding costs between \$20,000 and \$30,000: **0.4709**
- **(c)** To be in the 5% most expensive weddings, it must cost at least **\$39,060**

---

Would you like further details or explanations on any part?  
Here are 5 related questions:

1. What is the probability that a wedding costs more than \$40,000?
2. How would the probability change if the standard deviation increased?
3. What is the Z-score for a wedding cost of \$35,000?
4. How would you calculate the interquartile range (IQR) for wedding costs?
5. What does the Z-score tell you about a wedding cost compared to the average?

**Tip**: When using the normal distribution, always ensure that the mean and standard deviation are correctly interpreted from the context for accurate Z-score calculation.

The XO Group Inc. conducted a 2015 survey of 13000 brides and grooms married in the United States and found that the average cost of a wedding is $29858 (XO Group website). Assume that the cost of a wedding is normally distributed with a mean of 29858 and a standard deviation of 5600. 

a. What is the probability that a wedding costs less than $20000 (to 4 decimals)? 

b. What is the probability that a wedding costs between 20000 and 30000 (to 4 decimals)? 

c. For a wedding to be among the 5% most expensive, how much would it have to cost (to the nearest whole number)?

Learn how to calculate the probability of wedding costs using normal distribution with a mean of $29,858 and standard deviation of $5,600. Solve for the probability that a wedding costs less than $20,000, between $20,000 and $30,000, and determine the cost to be in the top 5%.

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