Shelia's measured glucose level one hour after a sugary drink varies according to the normal distribution with μ = 117 mg/dl and σ = 10.6 mg/dl. What is the level L such that there is probability only 0.01 that the mean glucose level of 6 test results falls above L ? Give your answer precise to one decimal place.

There is a probability of 0.01 that the mean glucose level of 6 test results falls above 141.7 mg/dl

Z score is used to determine by how many standard deviations the raw score is above or below the mean. The z score is given by:

Given that mean(μ) = 117 mg/dl and standard deviation (σ) = 10.6 mg/dl

The z score that corresponds to an area to the right of 0.01 is 2.33.

Hence:

Hence There is a probability of 0.01 that the mean glucose level of 6 test results falls above 141.7 mg/dl

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The level L such that there is probability only 0.01 that the mean glucose level of 6 test results falls above L is L = 127.1 mg/dl.

Step-by-step explanation:

To solve this problem, we need 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 , a large sample size can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

What is the level L such that there is probability only 0.01 that the mean glucose level of 6 test results falls above L ?

This is the value of X when Z has a pvalue of 1-0.01 = 0.99. So X when Z = 2.33.

By the Central Limit Theorem

The level L such that there is probability only 0.01 that the mean glucose level of 6 test results falls above L is L = 127.1 mg/dl.