Statistics/probability problem 2

**jackiemoon** · January 2nd 2009, 02:13 PM

Serum albumin is an important blood protein. In a particular population of healty individuals the distribution of serum albumin is found to be Normal with mean 40 gL ^-1 and standard deviation 5 gL ^-1.

(a) What fraction of the population will have a serum albumin level less than 32?
(b) Clinicians use reference intervals to characterize the central 95% of the population: this interval is then used (together with other indices) to identify patients with exceptional blood characteristics. Calculate the reference interval for this population.
(c) If two healthy people are randomly selected and tested for serum albumin, what is the probability that they will give test scores more than 15 units apart?
(d) A clinician wants to use the sum of the serum albumin result (A) and that for another clinically interesting blood parameter (B), which is also Normally distributed, as a diagnostic index. The second score, B, has a mean of 10 and a standard deviation of 2 in the healthy population. The two scores, A and B, are correlated with a correlation of p = 0.2 [note; p = Cov(A,B)/[SD(A).SD(B)], where Cov is covariance]. What value Tc should she set such that Tc = A + B is greater than T for only 1% of the healthy population?
(e) How would you assess whether or not data were Normally distributed?

I'm very hazy about solving these problems and would appreciate any tips or solutions.

Thank you

**mr fantastic** · January 2nd 2009, 09:01 PM

Originally Posted by jackiemoon

Serum albumin is an important blood protein. In a particular population of healty individuals the distribution of serum albumin is found to be Normal with mean 40 gL ^-1 and standard deviation 5 gL ^-1.

(a) What fraction of the population will have a serum albumin level less than 32?
(b) Clinicians use reference intervals to characterize the central 95% of the population: this interval is then used (together with other indices) to identify patients with exceptional blood characteristics. Calculate the reference interval for this population.
(c) If two healthy people are randomly selected and tested for serum albumin, what is the probability that they will give test scores more than 15 units apart?
(d) A clinician wants to use the sum of the serum albumin result (A) and that for another clinically interesting blood parameter (B), which is also Normally distributed, as a diagnostic index. The second score, B, has a mean of 10 and a standard deviation of 2 in the healthy population. The two scores, A and B, are correlated with a correlation of p = 0.2 [note; p = Cov(A,B)/[SD(A).SD(B)], where Cov is covariance]. What value Tc should she set such that Tc = A + B is greater than T for only 1% of the healthy population?
(e) How would you assess whether or not data were Normally distributed?

I'm very hazy about solving these problems and would appreciate any tips or solutions.

Thank you

I only have time to get you started.

X ~ Normal $(\mu = 40, \sigma = 5)$ .

(a) Calculate $\Pr(X < 32)$ .

(b) Find the the value of $a$ such that $\Pr(X > a) = 0.025$ . Then the required interval is $(-a, a)$ .

Time permitting, I will post more later. In the meantime you should think harder about the other questions ....

**mr fantastic** · January 3rd 2009, 12:49 AM

Originally Posted by jackiemoon

Serum albumin is an important blood protein. In a particular population of healty individuals the distribution of serum albumin is found to be Normal with mean 40 gL ^-1 and standard deviation 5 gL ^-1.

(a) What fraction of the population will have a serum albumin level less than 32?
(b) Clinicians use reference intervals to characterize the central 95% of the population: this interval is then used (together with other indices) to identify patients with exceptional blood characteristics. Calculate the reference interval for this population.
(c) If two healthy people are randomly selected and tested for serum albumin, what is the probability that they will give test scores more than 15 units apart?
(d) A clinician wants to use the sum of the serum albumin result (A) and that for another clinically interesting blood parameter (B), which is also Normally distributed, as a diagnostic index. The second score, B, has a mean of 10 and a standard deviation of 2 in the healthy population. The two scores, A and B, are correlated with a correlation of p = 0.2 [note; p = Cov(A,B)/[SD(A).SD(B)], where Cov is covariance]. What value Tc should she set such that Tc = A + B is greater than T for only 1% of the healthy population?
(e) How would you assess whether or not data were Normally distributed?

I'm very hazy about solving these problems and would appreciate any tips or solutions.

Thank you

(c) Theorem: If $X_1$ and $X_2$ are two independent normal random variables with means $\mu_1$ and $\mu_2$ and variances $\sigma_1^2$ and $\sigma_2^2$ respectively then the random variable $U = X_1 - X_2$ is also a normal random variable with mean $\mu_1 - \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ :

$U = X_1 - X_2$ ~ Normal $(\mu = \mu_1 - \mu_2, \, \sigma^2 = \sigma_1^2 + \sigma_2^2)$ .

So consider $U$ ~ Normal $(\mu = 0, \, \sigma^2 = 5^2 + 5^2 = 50)$ and calculate $\Pr(U > 15) + \Pr(U < -15) = 2 \Pr(U > 15)$ .

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