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The mean per capita consumption of milk per year is 131 liters with a variance of 841. If a sample of 132 people is randomly selected, what is the probability that the sample mean would be less than 133.5 liters

User Hearner
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1 Answer

3 votes

Answer:

0.8389 = 83.89% probability that the sample mean would be less than 133.5 liters.

Explanation:

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

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
\mu and standard deviation
\sigma, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
\mu and standard deviation
\sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
\mu and standard deviation
s = (\sigma)/(√(n)).

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The mean per capita consumption of milk per year is 131 liters with a variance of 841.

This means that
\mu = 131, \sigma = √(841) = 29

Sample of 132 people

This means that
n = 132, s = (29)/(√(132))

What is the probability that the sample mean would be less than 133.5 liters?

This is the p-value of Z when X = 133.5. So


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

By the Central Limit Theorem


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


Z = (133.5 - 131)/((29)/(√(132)))


Z = 0.99


Z = 0.99 has a p-value of 0.8389

0.8389 = 83.89% probability that the sample mean would be less than 133.5 liters.

User Maddy
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