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A distribution of values is normal with a mean of 60 and a standard deviation of 16. From this distribution, you are drawing samples of size 25. Find the interval containing the middle-most 76% of sample means.

User Alaskan
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Answer:

The interval containing the middle-most 76% of sample means is between 56.24 and 63.76.

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.

A distribution of values is normal with a mean of 60 and a standard deviation of 16.

This means that
\mu = 60, \sigma = 16

Samples of size 25:

This means that
n = 25, s = (16)/(√(25)) = 3.2

Find the interval containing the middle-most 76% of sample means.

Between the 50 - (76/2) = 12th percentile and the 50 + (76/2) = 88th percentile.

12th percentile:

X when Z has a p-value of 0.12, so X when Z = -1.175.


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

By the Central Limit Theorem


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


-1.175 = (X - 60)/(3.2)


X - 60 = -1.175*3.2


X = 56.24

88th percentile:


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


1.175 = (X - 60)/(3.2)


X - 60 = 1.175*3.2


X = 63.76

The interval containing the middle-most 76% of sample means is between 56.24 and 63.76.

User Sgcharlie
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