Question

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.)

Answer 1

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 .

Answer 2

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.