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Judy’s measured potassium level varies according to the Normal distribution with μ = 3.8 and σ = 0.2 mmol/l. Let us consider what could happen if we took four separate measurements from Judy.

What is the blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L? Hint: This requires an inverse Normal calculation. (Enter you answer rounded to two decimal places.)

2 Answers

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Answer:

Blood potassium level L = 3.64

Explanation:

We are given that Judy’s measured potassium level varies according to the Normal distribution with μ = 3.8 and σ = 0.2 mm .

Let X bar = average of four measurements

So, X bar ~ N(
\mu = 3.8,\sigma = 0.2)

The z score probability distribution of sample mean is ;

Z =
(Xbar -\mu)/((\sigma)/(√(n) ) ) ~ N(0,1)

where, n = sample size = 4


\mu = population mean = 3.8


\sigma = population standard deviation = 0.2

The condition given to us is that probability that the average of four measurements is less than L is only 0.05 , i.e.;

P(X bar < L) = 0.05

P(
(Xbar -\mu)/((\sigma)/(√(n) ) ) <
(L -3.8)/((0.2)/(√(4) ) ) ) = 0.05

P(Z <
(L -3.8)/(0.1 ) ) = 0.05

In z table, we find that the critical value below which the z probability is 0.05 is given as -1.6449 .

So, it means;
(L -3.8)/(0.1 ) = -1.6449

L = (-1.6449 * 0.1) + 3.8

L = -0.16449 + 3.8 = 3.64

Therefore, the blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L is 3.64 .

User Mattis
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3 votes

Answer:

The blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L is 3.64.

Explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
\mu and standard deviation
\sigma, the zscore of a measure X is given by:


Z = (X - \mu)/(\sigma)

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
\mu and standard deviation
\sigma, the sample means of size n can be approximated to a normal distribution with mean
\mu and standard deviation, which is also called standard error
s = (\sigma)/(√(n))

In this problem, we have that:


\mu = 3.8, \sigma = 0.2, n = 4, s = (0.2)/(√(4)) = 0.1

What is the blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L?

This is the value of X when Z has a pvalue of 0.05. So it is X when Z = -1.645.


Z = (X - \mu)/(\sigma)

By the Central Limit Theorem


Z = (X - \mu)/(s)


-1.645 = (X - 3.8)/(0.1)


X - 3.8 = -1.645*0.1


X = 3.64

The blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L is 3.64.

User Dmitrii Erokhin
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7.5k points