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madam [21]
3 years ago
6

If a normally distributed population has a mean (mu) that equals 100 with a standard deviation (sigma) of 18, what will be the c

omputed z-score with a sample mean (x-bar) of 106 from a sample size of 9?
Mathematics
1 answer:
garik1379 [7]3 years ago
7 0

Answer:

Z = 1

Step-by-step 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 = \frac{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 = \frac{\sigma}{\sqrt{n}}.

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

Mean (mu) that equals 100 with a standard deviation (sigma) of 18

\mu = 100, \sigma = 18

Sample of 9:

This means that n = 9, s = \frac{18}{\sqrt{9}} = 6

What will be the computed z-score with a sample mean (x-bar) of 106?

This is Z when X = 106. So

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{106 - 100}{6}

Z = 1

So Z = 1 is the answer.

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