Respuesta :
Answer:
(a) 0.04006
(b) 0.12924
(c) 0.29116
(d) $225.76
Step-by-step explanation:
We are given that Money magazine reported that a visit to a hospital emergency room for something as simple as a sore throat has a mean cost of $348 with a standard deviation of $87.
Let X = cost of a hospital emergency room visit for this medical service
So, X ~ N([tex]\mu=348, \sigma^{2} =87^{2}[/tex])
The standard normal z score distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
(a) Probability that the cost will be more than $500 = P(X > $500)
P(X > 500) = P( [tex]\frac{X-\mu}{\sigma}[/tex] > [tex]\frac{500-348}{87}[/tex] ) = P(Z > 1.75) = 1 - P(Z <= 1.75)
= 1 - 0.95994 = 0.04006
(b) Probability that the cost will be less than $250 = P(X < $250)
P(X < 250) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{250-348}{87}[/tex] ) = P(Z < -1.13) = 1 - P(Z <= 1.13)
= 1 - 0.87076 = 0.12924
(c) Probability that the cost will be between $300 and $400 = P(300<X<400)
P($300 < X < $400) = P(X < $400) - P(X <= $300)
P(X < 400) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{400-348}{87}[/tex] ) = P(Z < 0.60) = 0.72575
P(X <= 300) = P( [tex]\frac{X-\mu}{\sigma}[/tex] <= [tex]\frac{300-348}{87}[/tex] ) = P(Z <= -0.55) = 1 - P(Z < 0.55)
= 1 - 0.70884 = 0.29116
Therefore, P($300 < X < $400) = 0.72575 - 0.29116 = 0.43459
(d) Now, it is given that the cost to a patient is in the lower 8% of charges for this medical service, i.e.;
P(Z < x) = 0.08 ⇒ P(Z < [tex]\frac{X-348}{87}[/tex] ) = 0.08
From Z table it is found that the z score has a probability of 8% at a critical value of x = -1.40507
which means, [tex]\frac{X-348}{87}[/tex] = -1.40507
X = (-1.40507 * 87) + 348 = $225.76
Therefore, the cost of this patient’s emergency room visit is $225.76 .
