In an article about the cost of health care, Money magazine...

In an article about the cost of health care, Money magazine reported that a visit to a hospital emergency room for something as simple as a sore throat has a mean cost of $\$ 328$ (Money, January 2009 ). Assume that the cost for this type of hospital emergency room visit is normally distributed with a standard deviation of $\$ 92 .$ Answer the following questions about the cost of a hospital emergency room visit for this medical service. a. What is the probability that the cost will be more than $\$ 500 ?$ b. What is the probability that the cost will be less than $\$ 250 ?$ c. What is the probability that the cost will be between $\$ 300$ and $\$ 400$ ? d. If the cost to a patient is in the lower 8$\%$ of charges for this medical service, what was the cost of this patient's emergency room visit?

In an article about the cost of health care, Money magazine reported that a visit to a hospital emergency room for something as simple as a sore throat has a mean cost of $\$ 328$
(Money, January 2009 ). Assume that the cost for this type of hospital emergency room visit
is normally distributed with a standard deviation of $\$ 92 .$ Answer the following questions
about the cost of a hospital emergency room visit for this medical service.
a. What is the probability that the cost will be more than $\$ 500 ?$
b. What is the probability that the cost will be less than $\$ 250 ?$
c. What is the probability that the cost will be between $\$ 300$ and $\$ 400$ ?
d. If the cost to a patient is in the lower 8$\%$ of charges for this medical service, what was
the cost of this patient's emergency room visit?

Key Concepts

Z-Score Transformation

A z-score is a standardized score that represents how many standard deviations an element is from the mean. By converting a value to a z-score, one can use the standard normal distribution to compute probabilities. This transformation is key in comparing different normal distributions and solving probability problems.

Standard Deviation

Standard deviation is a measure of the dispersion or spread of a set of values in a distribution. It indicates how much the individual data points deviate from the mean on average. In a normal distribution, about 68% of the data falls within one standard deviation from the mean, which aids in interpreting probabilities.

Percentile and Inverse Probability Computation

Percentiles indicate the relative standing of a value within a distribution, showing the percentage of data below that value. Inverse probability computation involves determining the value corresponding to a given cumulative probability (or percentile) in the distribution. This process typically requires the use of z-tables or statistical software to translate between probabilities and raw scores.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, characterized by its bell-shaped curve. It is widely used in statistics to represent real-valued random variables whose distributions are not known. This distribution is fully described by two parameters: the mean (which determines the location) and the standard deviation (which determines the spread).

Mean

The mean of a distribution is the average value around which the data tends to cluster. In the context of the normal distribution, the mean is the center of symmetry of the bell-shaped curve, and it represents the expected value of the random variable.

Transcript

00:01 All right, so we're given that it was reported that the cost for an hospital emergency room for something as simple as a short throat has a mean cost of $328 .28.

00:20 And we're assuming that the cost is normally distributed with a standard deviation of $92.

00:28 Part a.

00:30 First off, let's assign the random variable x equal cost of high.

00:38 Hospital visit for something as simple as a sore throat, yada, yada, yada.

00:47 Part a, we need to find the probability that the cost will be more than $500.

00:57 First off, let's z score this 500.

01:00 So z equals 500 and is 328, all over 92.

01:12 You get a z score of 1 .87, and then you get that the probability that z is less than or equal to 1 .87 is looking at our table.

01:29 Sorry, 0 .9663.

01:34 Sorry, 9393.

01:38 Okay, yeah, i'm just better off deleting this.

01:41 There we go all right so but this right here is the probability that z is greater than 1 .97 so we need 1 minus the probability and z is less than or equal to 1 .87 so 1 minus 0 .9693 and you get that this equals 0 .0307 part b we need to find the probability that the cost is less than $250 all right all right, let's do our z scoring.

02:25 So z equals 250, it's 328, all over 92.

02:36 Quickly punching this into a calculator, we get that the z score is equal to negative 0 .85.

02:44 And we get looking at our chart, which i need to flip over because, yeah, we get the probability that to the left of this is equal to to 0 .1, 2, 3, 4, 5.

03:03 Okay, 0 .1977.

03:07 So this is the probability that z is less than, well, technically less than.

03:15 It doesn't matter in the grand scheme of things.

03:21 0 .1977.

03:24 All right.

03:26 Next page.

03:27 Rewriding info for the sake of not having to cross -reference different pages.

03:37 All right, part c, this gives us a range to check.

03:44 So we need to check that the probability of x is between $300 and $400.

03:55 All right, so let's find our two z scores.

03:58 I'm going to denote it z sub l and z sub u for z score lower and z score upper...