Answered

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 :

Answer:

a) 0.0392

b) 0.4688

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

Step-by-step explanation:

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

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

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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:

[tex]\mu = 29858, \sigma = 5600[/tex]

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

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{20000 - 29858}{5600}[/tex]

[tex]Z = -1.76[/tex]

[tex]Z = -1.76[/tex] 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

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{30000 - 29858}{5600}[/tex]

[tex]Z = 0.02[/tex]

[tex]Z = 0.02[/tex] has a pvalue of 0.5080.

X = 20000

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{20000 - 29858}{5600}[/tex]

[tex]Z = -1.76[/tex]

[tex]Z = -1.76[/tex] 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.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]1.645 = \frac{X - 29858}{5600}[/tex]

[tex]X - 29858 = 5600*1.645[/tex]

[tex]X = 39070[/tex]

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

Other Questions