Math Problem Statement

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)?

Solution

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=xμσz = \frac{x - \mu}{\sigma}

where:

  • xx is the value you're analyzing.
  • μ=29,858\mu = 29,858 is the mean.
  • σ=5,600\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=20,00029,8585,600z = \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:

    z1=20,00029,8585,600z_1 = \frac{20,000 - 29,858}{5,600} z2=30,00029,8585,600z_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 z1z_1 from that of z2z_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=zσ+μx = 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.

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Math Problem Analysis

Mathematical Concepts

Normal Distribution
Probability
Z-score

Formulas

z = (x - μ) / σ

Theorems

Z-score calculation for normal distribution

Suitable Grade Level

Grades 11-12 or college-level statistics