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Let's break down the problem step by step. The information provided:

- The average (mean) cost of a wedding is $29,858.
- The standard deviation is $5,600.
- The distribution of wedding costs is assumed to be normally distributed.

We will solve each part of the problem using the z-score formula for normal distribution:

\[
z = \frac{x - \mu}{\sigma}
\]
where:
- $x$ is the value you're analyzing.
- $\mu = 29,858$ is the mean.
- $\sigma = 5,600$ is the standard deviation.

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

1. Calculate the z-score for $20,000:
   \[
   z = \frac{20,000 - 29,858}{5,600}
   \]

2. Using the z-score, we can then find the probability (area to the left of this z-score) using the standard normal distribution table.

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

1. Calculate the z-scores for both $20,000 and $30,000:
   \[
   z_1 = \frac{20,000 - 29,858}{5,600}
   \]
   \[
   z_2 = \frac{30,000 - 29,858}{5,600}
   \]

2. The probability will be the area between these two z-scores, which can be found by subtracting the cumulative probability of $z_1$ from that of $z_2$.

### Part (c) - For a wedding to be among the 5% most expensive:

1. We need to find the cost corresponding to the 95th percentile, as only 5% of weddings are more expensive than this value.
   
2. The z-score corresponding to the 95th percentile is approximately 1.645. Using the z-score formula, we can solve for the cost:
   \[
   x = z \cdot \sigma + \mu
   \]

Let me calculate each of these steps.Here are the results for each part:

### Part (a)
The probability that a wedding costs less than $20,000 is approximately **0.0392** (or 3.92%).

### Part (b)
The probability that a wedding costs between $20,000 and $30,000 is approximately **0.4709** (or 47.09%).

### Part (c)
For a wedding to be among the 5% most expensive, it would need to cost at least **$39,069**.

Would you like further details or have any questions?

Here are some related questions to explore:
1. How would the results change if the standard deviation were smaller?
2. What is the probability that a wedding costs more than $35,000?
3. How would you calculate the top 1% most expensive weddings?
4. What percentage of weddings cost between $25,000 and $35,000?
5. How would the mean affect the results if it were higher?

**Tip**: Understanding z-scores is essential when dealing with normal distributions, as they standardize data to compare different values.

The XO Group Inc. conducted a 2015 survey of 13,000 brides and grooms married in the United States and found that the average cost of a wedding is $29,858. Assume that the cost of a wedding is normally distributed with a mean of $29,858 and a standard deviation of $5,600. a) What is the probability that a wedding costs less than $20,000 (to 4 decimals)? b) What is the probability that a wedding costs between $20,000 and $30,000 (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)?

This problem involves calculating probabilities using normal distribution for wedding costs based on a 2015 survey. It covers the Z-score formula and calculating costs that fall below $20,000, between $20,000 and $30,000, and the top 5% of most expensive weddings. Ideal for students in Grades 11-12 or introductory college-level statistics.

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