Question
The XO Group Inc. conducted a 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 cost between $$20,000 and $30,000 (to 4 decimals)?
c. For a wedding to be among the 5% most expensive, how much would it have to cost ( to the nearest whole number)?
a. What is the probability that a wedding costs less than $20,000 (to 4 decimals)?
b. what is the probability that a wedding cost between $$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)?
Answers
GPT 3.5
To solve these problems, we need to use the z-score formula for normally distributed data:
z = (x - μ) / σ
Where:
z = z-score
x = value we want to find the probability for
μ = mean of the data
σ = standard deviation of the data
a. To find the probability that a wedding costs less than $20,000:
z = (20000 - 29858) / 5600 = -1.5389 (to 4 decimals)
Using a z-table or calculator, we can find that the probability corresponding to a z-score of -1.5389 is approximately 0.0611 (to 4 decimals).
b. To find the probability that a wedding costs between $20,000 and $30,000:
We need to find the cumulative probability for both values and subtract them:
For $20,000:
z = (20000 - 29858) / 5600 = -1.5389 (same as a.)
Probability at z = -1.5389 is 0.0611 (from a.)
For $30,000:
z = (30000 - 29858) / 5600 = 0.2554 (to 4 decimals)
Using a z-table or calculator, we can find that the probability corresponding to a z-score of 0.2554 is approximately 0.5988 (to 4 decimals).
Therefore, the probability of a wedding costing between $20,000 and $30,000 is 0.5988 - 0.0611 = 0.5377 (to 4 decimals).
c. To find the cost of a wedding that is among the 5% most expensive, we need to find the z-score that corresponds to a cumulative probability of 0.95 (1 - 0.05).
Using a z-table or calculator, we find that the z-score corresponding to a cumulative probability of 0.95 is approximately 1.645 (to 3 decimals).
Now we can solve for x, using the z-score formula:
1.645 = (x - 29858) / 5600
Solving for x:
1.645 * 5600 = x - 29858
9184.2 = x - 29858
x = 39042.2
Therefore, a wedding would need to cost approximately $39,042 (to the nearest whole number) to be among the 5% most expensive.
z = (x - μ) / σ
Where:
z = z-score
x = value we want to find the probability for
μ = mean of the data
σ = standard deviation of the data
a. To find the probability that a wedding costs less than $20,000:
z = (20000 - 29858) / 5600 = -1.5389 (to 4 decimals)
Using a z-table or calculator, we can find that the probability corresponding to a z-score of -1.5389 is approximately 0.0611 (to 4 decimals).
b. To find the probability that a wedding costs between $20,000 and $30,000:
We need to find the cumulative probability for both values and subtract them:
For $20,000:
z = (20000 - 29858) / 5600 = -1.5389 (same as a.)
Probability at z = -1.5389 is 0.0611 (from a.)
For $30,000:
z = (30000 - 29858) / 5600 = 0.2554 (to 4 decimals)
Using a z-table or calculator, we can find that the probability corresponding to a z-score of 0.2554 is approximately 0.5988 (to 4 decimals).
Therefore, the probability of a wedding costing between $20,000 and $30,000 is 0.5988 - 0.0611 = 0.5377 (to 4 decimals).
c. To find the cost of a wedding that is among the 5% most expensive, we need to find the z-score that corresponds to a cumulative probability of 0.95 (1 - 0.05).
Using a z-table or calculator, we find that the z-score corresponding to a cumulative probability of 0.95 is approximately 1.645 (to 3 decimals).
Now we can solve for x, using the z-score formula:
1.645 = (x - 29858) / 5600
Solving for x:
1.645 * 5600 = x - 29858
9184.2 = x - 29858
x = 39042.2
Therefore, a wedding would need to cost approximately $39,042 (to the nearest whole number) to be among the 5% most expensive.