Respuesta :
Answer:
(a) The probability that the wedding costs more than $31,000 is 0.0427.
(b) The probability that the wedding costs between $26,000 and $30,000 is 0.8005.
(c) The probability that the wedding costs less than $25,000 is 0.0113.
Step-by-step explanation:
Let X = cost of a wedding.
The random variable X follows a Normal distribution with mean, μ = $28, 423 and the standard deviation, σ = $1,500.
(a)
Compute the probability that the wedding costs more than $31,000 as follows:
[tex]P(X>31000)=P(\frac{X-\mu}{\sigma}>\frac{31000-28423}{1500})\\=P(Z>1.72)\\=1-P(Z<1.72)\\=1-0.9573\\=0.0427[/tex]
*Use a z-table for the probability.
Thus, the probability that the wedding costs more than $31,000 is 0.0427.
(b)
Compute the probability that the wedding costs between $26,000 and $30,000 as follows:
[tex]P(26000<X<30000)=P(\frac{26000-28423}{1500}<\frac{X-\mu}{\sigma}<\frac{30000-28423}{1500})\\=P(-1.62<Z<1.05)\\=P(Z<1.05)-P(Z<-1.62)\\=0.8531-0.0526\\=0.8005[/tex]
*Use a z-table for the probability.
Thus, the probability that the wedding costs between $26,000 and $30,000 is 0.8005.
(c)
Compute the probability that the wedding costs less than $25,000 as follows:
[tex]P(X<25000)=P(\frac{X-\mu}{\sigma}<\frac{25000-28423}{1500})\\=P(Z<-2.28)\\=1-P(Z<2.28)\\=1-0.9887\\=0.0113[/tex]
*Use a z-table for the probability.
Thus, the probability that the wedding costs less than $25,000 is 0.0113.
Answer:
(a) P(X > $31,000) = 0.0427
(b) P($26,000 < X < $30,000) = 0.8005
(c) P(X < $25,000) = 0.0113
Step-by-step explanation:
We are given that according to an annual survey conducted by a particular website, the mean cost of a wedding in 2012 was $28,423. Suppose the cost for a wedding is normally distributed, with a standard deviation of $1,500, and a wedding is selected at random.
Let X = cost for a wedding
So, X ~ N([tex]\mu=$28423,\sigma^{2} = $1,500^{2}[/tex])
The z score probability distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
(a) Probability that the wedding costs more than $31,000 = P(X>$31,000)
P(X > 31,000) = P( [tex]\frac{X-\mu}{\sigma}[/tex] > [tex]\frac{31,000-28,423}{1,500}[/tex] ) = P(Z > 1.72) = 1 - P(Z <= 1.72)
= 1 - 0.95728 = 0.0427
(b) Probability that the wedding costs between $26,000 and $30,000 = P($26,000 < X < $30,000) = P(X < $30,000) - P(X <= $26,000) \
P(X < 30,000) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{30,000-28,423}{1,500}[/tex] ) = P(Z < 1.05) = 0.85314
P(X <= 26,000) = P( [tex]\frac{X-\mu}{\sigma}[/tex] <= [tex]\frac{26,000-28,423}{1,500}[/tex] ) = P(Z <= -1.62) = 1 - P(Z <= 1.62)
= 1 - 0.94738 = 0.05262
Therefore, P($26,000 < X < $30,000) = 0.85314 - 0.05262 = 0.8005 .
(c) Probability that the wedding costs less than $25,000 = P(X < $25,000)
P(X < 25,000) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{25,000-28,423}{1,500}[/tex] ) = P(Z < -2.28) = 1 - P(Z <= 2.28)
= 1 - 0.98870 = 0.0113
