Math Problem Statement
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:
where:
- is the value you're analyzing.
- is the mean.
- is the standard deviation.
Part (a) - Probability that a wedding costs less than $20,000:
-
Calculate the z-score for $20,000:
-
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:
-
Calculate the z-scores for both $20,000 and $30,000:
-
The probability will be the area between these two z-scores, which can be found by subtracting the cumulative probability of from that of .
Part (c) - For a wedding to be among the 5% most expensive:
-
We need to find the cost corresponding to the 95th percentile, as only 5% of weddings are more expensive than this value.
-
The z-score corresponding to the 95th percentile is approximately 1.645. Using the z-score formula, we can solve for the cost:
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:
- How would the results change if the standard deviation were smaller?
- What is the probability that a wedding costs more than $35,000?
- How would you calculate the top 1% most expensive weddings?
- What percentage of weddings cost between $25,000 and $35,000?
- 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