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 (XO Group website, January 5, 2015). Assume that the cost of a wedding is normally distributed with a mean of $29,858 and a standard deviation of $5600.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)?

Answer 1

a) 0.0392

b) 0.4688

c) At least $39,070 to be among the 5% most expensive.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 29858, \sigma = 5600`

a. What is the probability that a wedding costs less than $20,000 (to 4 decimals)?

This is the pvalue of Z when X = 20000. So

`Z = (X - \mu)/(\sigma)`

`Z = (20000 - 29858)/(5600)`

`Z = -1.76`

`Z = -1.76` has a pvalue of 0.0392.

So this probability is 0.0392.

b. What is the probability that a wedding costs between $20,000 and $30,000 (to 4 decimals)?

This is the pvalue of Z when X = 30000 subtracted by the pvalue of Z when X = 20000.

X = 30000

`Z = (X - \mu)/(\sigma)`

`Z = (30000 - 29858)/(5600)`

`Z = 0.02`

`Z = 0.02` has a pvalue of 0.5080.

X = 20000

`Z = (X - \mu)/(\sigma)`

`Z = (20000 - 29858)/(5600)`

`Z = -1.76`

`Z = -1.76` has a pvalue of 0.0392.

So this probability is 0.5080 - 0.0392 = 0.4688

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

This is the value of X when Z has a pvalue of 0.95. So this is X when Z = 1.645.

`Z = (X - \mu)/(\sigma)`

`1.645 = (X - 29858)/(5600)`

`X - 29858 = 5600*1.645`

`X = 39070`

The wedding would have to cost at least $39,070 to be among the 5% most expensive.