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At the conclusion of training, all factory workers are timed as they complete a task. The completion times for the task are normally distributed.


Suppose a worker's completion time is 1.1 standard deviations above the mean. What is the worker's percentile score? Round your result to one decimal place.


The annual salaries for people in a particular profession are known to be normally distributed with a mean of $54,800 and a standard deviation of $2,600.

What percentage of people in this profession earn annual salaries between $54,000 and $57,000? Round your result to one decimal place.

Sagot :

Using the normal distribution, it is found that:

  • The worker's score is at the 86.4th percentile.
  • 42.2% of people in this profession earn annual salaries between $54,000 and $57,000.

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

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

  • 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.

Question 1:

1.1 standard deviations above the mean, hence, a z-score of Z = 1.1.

  • Looking at the z-table, z = 1.1 has a p-value of 0.864, hence, the worker's score is at the 86.4th percentile.

Question 2:

  • Mean of 54800, hence [tex]\mu = 54800[/tex]
  • Standard deviation of 2600, hence [tex]\sigma = 2600[/tex].

The proportion is the p-value of Z when X = 57000 subtracted by the p-value of Z when X = 54000, then:

X = 57000:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{57000 - 54800}{2600}[/tex]

[tex]Z = 0.846[/tex]

[tex]Z = 0.846[/tex] has a p-value of 0.801.

X = 54000:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{54000 - 54800}{2600}[/tex]

[tex]Z = -0.307[/tex]

[tex]Z = -0.307[/tex] has a p-value of 0.379.

0.801 - 0.379 = 0.422

0.422 x 100% = 42.2%

42.2% of people in this profession earn annual salaries between $54,000 and $57,000.

A similar problem is given at https://brainly.com/question/24663213

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