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.)

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.

Step-by-step 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 and standard deviation , the zscore of a measure X is given by:

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 and standard deviation , the sample means of size n can be approximated to a normal distribution with mean and standard deviation, which is also called standard error

In this problem, we have that:

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.

By the Central Limit Theorem

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.

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