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We take a sample of size n=16 for a quality of life (QoL) measurement from a population with a normal distribution with m=8 and s=6.

a) What is the expected mean and the standard error for a sample size of n=16

b) What is the probability that the sample mean will be greater than 10?

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Answer:

a) The mean is 8 and the standard error is 1.5.

b) 9.18% probability that the sample mean will be greater than 10.

Explanation:

To solve this question, it is important to know the normal probability distribution and the Central Limit Theorem.

Normal probability distribution

Problems of normally distributed samples can be 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, a large sample size can be approximated to a normal distribution with mean
\mu and standard deviation
(\sigma)/(√(n))

In this problem, we have that:

I call the population mean
\mu and the population standard deviation
\sigma

So


\mu = 8, \sigma = 6

a) What is the expected mean and the standard error for a sample size of n=16

By the Central Limit Theorem


\mu = 8


s = (6)/(√(16)) = 1.5

b) What is the probability that the sample mean will be greater than 10?

This probability is 1 subtracted by the pvalue of Z when X = 10.


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

Due to the Central Limit Theorem


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


Z = (10 - 8)/(1.5)


Z = 1.33


Z = 1.33 has a pvalue of 0.9082.

So there is a 1-0.9082 = 0.0918 = 9.18% probability that the sample mean will be greater than 10.

User Paul Mendoza
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