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The height of women ages​ 20-29 is normally​ distributed, with a mean of 64.9 inches. Assume sigmaequals2.3 inches. Are you more likely to randomly select 1 woman with a height less than 66 inches or are you more likely to select a sample of 27 women with a mean height less than 66 ​inches? Explain.

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Answer:

To select a sample of 27 women with a mean height less than 66 ​inches is more likely than to randomly select 1 woman with a height less than 66 inches

Step-by-step explanation:

1) The probability of randomly selecting 1 woman with a height less than 66 inches is

P(z<z(66)) where z(66) is the z-score of the woman whose height is 66 inches.

z score can be calculated using the formula

z(66)=[tex]\frac{X-M}{s}[/tex] where

  • X =66 inches
  • M is the mean height of women aged​ 20-29 (64.9 inches)
  • s is the standard deviation (2.3 inches)

Then z(66)=[tex]\frac{66-64.9}{2.3}[/tex] ≈ 0.48

and P(z<0.48) = 0.6844

2) The probability of selecting a sample of 27 women with a mean height less than 66 ​inches can be calculated using the equation

t=[tex]\frac{X-M}{\frac{s}{\sqrt{N} } }[/tex] where

  • X = 66 inches
  • M is the average height of women aged 20-29 (64.9 inches)
  • s is the standard deviation (2.3 inches)
  • N is the sample size (27)

t=[tex]\frac{66-64.9}{\frac{2.3}{\sqrt{27} } }[/tex] ≈ 2.49

looking t-table P(t<2.49)≈0.9903

Since 0.9903>0.6844 we can conclude that to select a sample of 27 women with a mean height less than 66 ​inches is more likely than to randomly select 1 woman with a height less than 66 inches

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