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A population has a mean of μ = 80 and a standard deviation of σ = 20. would a score of x = 70 be considered an extreme value (out in the tail) in this sample? if the standard deviation were σ = 5, would a score of x = 70 be considered an extreme value?

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DeanR

With a sigma of 20, a score of 70 is half a standard deviation below the mean 80,  z=-1/2.    I don't know if there's an official definition, but I consider values extreme if the absolute values their z scores is say 3 or more.  So no, this is not extreme at all.

If the standard deviation is 5 instead of 20, then 70 is 2 standard deviations below the mean (z = -2).  I still wouldn't consider that an extreme value.

Using the z-score, it is found that:

  • In the distribution, a score of x = 70 is not extreme.
  • If the distribution had a standard deviation of 5, it would be extreme.

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  • The z-score of a measure X in a distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:

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

  • It measures how many standard deviations X is from the mean.
  • If [tex]|Z| \geq 2[/tex], the measure X is considered to be extreme.

In this distribution:

  • Mean of 80, thus [tex]\mu = 80[/tex]
  • Standard deviation of 20, thus [tex]\sigma = 20[/tex]
  • Measure of [tex]X = 70[/tex]

The z-score is:

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

[tex]Z = \frac{70 - 80}{20}[/tex]

[tex]Z = -0.5[/tex]

  • [tex]|Z| = 0.5 < 2[/tex], thus, a score of x = 70 is not extreme.

With a standard deviation of [tex]\sigma = 5[/tex]

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

[tex]Z = \frac{70 - 80}{5}[/tex]

[tex]Z = -2[/tex]

  • [tex]|Z| = 2[/tex], thus, a score of x = 70 is extreme.

A similar problem is given at https://brainly.com/question/2908781

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