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Category: Calculus and Above
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# Create an example probability distribution to the right in

### Resolved Question:

Create an example probability distribution to the right in which the expected value is 15. You must have at least three different values for x.

What are the values for the mean and standard deviation of a standard normal distribution?
What are considered unusual z-score values?

The serum cholesterol levels in milligrams/deciliter (mg/dL) in a certain Mediterranean population are found to be normally distributed with a mean of 160 and a standard deviation of 50. Answer the following:

(a) Determine the z-score for a person from this population that has a cholesterol level of 115. Then find the z-score for someone whose cholesterol level is 242.
(b) If x represents a possible cholesterol level from this population, find P(x > 145).

(c) Find P(100 < x < 200) and give an interpretation of this value.

(d) The top 3% of all people in this group have cholesterol levels that make them "at-risk" for heart problems. Determine the raw-score cholesterol level which separates the at-risk people from the rest of the group.

Does a confidence interval for µ get wider or narrower if:
(a) the percent of desired confidence (confidence level) decreases from 99% to 90%?

(b) the size of the sample used to produce the confidence interval is decreased?
(c) our estimate of the standard deviation gets smaller?
Submitted: 1 year ago.
Category: Calculus and Above
Expert:  Chirag replied 1 year ago.
Hi, Welcome and Thanks for using Just Answer

I will assist you with these. I am working on this and will post the answers as soon as ready.

Thanks.
Customer: replied 1 year ago.

ok... do you have an idea of how long it may take?

Expert:  Chirag replied 1 year ago.
It may take a maximum of one hour. Is that ok?
Customer: replied 1 year ago.

That's fine, just need it in no more than an hour and a half

Expert:  Chirag replied 1 year ago.
Sure.
Expert:  Chirag replied 1 year ago.

(1)

Let the x- values be 3, 15, 25, 23 and the corresponding probabilities be 0.3, 0.3, 0.2, 0.2

The probability distribution is
x P(x)
3 0.3
15 0.3
25 0.2
23 0.2

(2)

Mean = 0 and Standard deviation = 1
z- scores less than -2 or greater than 2 are considered unusual

(3)

z = (x – μ)/σ
(a) For x = 115, z = (115 – 160)/50 = -0.90 and
for x = 242, z = (242 – 160)/50 = 1.64

(b) For x = 145, z = (145 – 160)/50 = -0.30
P(x > 145) = P(z > -0.30) = 0.618

(c) For x = 100, z = (100 – 160)/50 = -1.2 and
for x = 200, z = (200 – 160)/50 = 0.80
P(100 < x < 200) = P(-1.2 < z < 0.8) = 0.673
This means the probability of finding a person from the population with cholesterol level between 100 and 200 is 0.673

(d) The upper-tail z- score for 3% (0.03) is 1.881
x = μ + z * σ = 160 + 1.881 * 50 = 254.05

(4)

(a) The confidence interval gets narrower if the confidence level is decreased from 99% to 90%

(b) The confidence interval gets wider if the sample size is decreased

(c) The confidence interval gets narrower if the standard deviation gets smaller

Thanks.
Chirag, Master's Degree
Satisfied Customers: 12191
Experience: I have a Master's degree in Engineering and a very wide teaching experience of more than 25 years at various levels.
Customer: replied 1 year ago.

could you elaborate on 1? How did you get those values?

Expert:  Chirag replied 1 year ago.
(1)

We want the expected value to be 15. The expected value is given by the formula ∑(x * P(x)). So, we choose four values in such a way that when they are multiplied with the respective assumed probabilities, and the products added, the result should be 15. I did that and got four values of x as 3, 15, 25 and 23.

(2)

The normal distribution is defined by two parameters, μ and σ.
For the standard normal distribution, the values of μ and σ are respectively μ = 0 and σ = 1 (These are standardized values, and are fixed).

Area under the normal curve gives the probabilities. Area to the left of z = -2 is 0.0228 and area to the right of z = 2 is also 0.0228. This means the probability is just 2.3% for z- values less than -2 or greater than 2. Since this probability is low, z 2 is considered unusual range.

Thanks.

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