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Imee · Answer 1 · 2022-08-16T09:11:33+0000

Answer:

0% probability that the mean of the sample taken is less than 2.2 feet.

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.

Mean of 2.5 feet and a standard deviation of 0.2 feet.

This means that
$\mu = 2.5, \sigma = 0.2$

Sample of 41

This means that
n = 41, s = (0.2)/(√(41))

Find the probability that the mean of the sample taken is less than 2.2 feet.

This is the p-value of Z when X = 2.2 So

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

By the Central Limit Theorem

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

Z = (2.2 - 2.5)/((0.2)/(√(41)))

Z = -9.6

Z = -9.6 has a p-value of 0.

0% probability that the mean of the sample taken is less than 2.2 feet.

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