Question

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

Answer 1

Using the normal distribution and the central limit theorem, 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.

Normal Probability Distribution

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 p-value of Z when X = 270 subtracted by the p-value of Z when X = 240, 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 normal distribution and the central limit theorem, you can check brainly.com/question/24663213