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sladkih [1.3K]
2 years ago
6

(1) The scores ou an aptitude test for finger dexterity are uonually distributed with meau 250 and

Mathematics
1 answer:
satela [25.4K]2 years ago
7 0

Using the <em>normal distribution and the central limit theorem</em>, it is found that there is a:

a) 0.1813 = 18.13% probability that a person selected at random will score between 240 and 270 on the test.

b) 0.4501 = 45.01% probability that the mean score for the sample will be between 240 and 270.

<h3>Normal Probability Distribution</h3>

In a normal distribution with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

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

  • It 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, which is the percentile of X.
  • By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation s = \frac{\sigma}{\sqrt{n}}.

In this problem:

  • The mean is of \mu = 250.
  • The standard deviation is of \sigma = 65.

Item a:

The probability is the <u>p-value of Z when X = 270 subtracted by the p-value of Z when X = 240</u>, hence:

X = 270:

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

Z = \frac{270 - 250}{65}

Z = 0.31

Z = 0.31 has a p-value of 0.6217.

X = 240:

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

Z = \frac{240 - 250}{65}

Z = -0.15

Z = -0.15 has a p-value of 0.4404.

0.6217 - 0.4404 = 0.1813.

0.1813 = 18.13% probability that a person selected at random will score between 240 and 270 on the test.

Item b:

We have a sample of 7, hence:

n = 7, s = \frac{65}{\sqrt{7}} = 24.57

X = 270:

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

By the Central Limit Theorem

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

Z = \frac{270 - 250}{24.57}

Z = 0.81

Z = 0.81 has a p-value of 0.7910.

X = 240:

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

Z = \frac{240 - 250}{24.57}

Z = -0.41

Z = -0.41 has a p-value of 0.3409.

0.7910 - 0.3409 = 0.4501.

0.4501 = 45.01% probability that the mean score for the sample will be between 240 and 270.

To learn more about the <em>normal distribution and the central limit theorem</em>, you can check brainly.com/question/24663213

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